Does Smooth Ambiguity Matter for Asset Pricing?

K.7 Does Smooth Ambiguity Matter for Asset Pricing? Gallant, A. Ronald, Mohammad R. Jahan-Parvar, and Hening Liu Please cite paper as: Gallant, A. Ronald, Mohammad R. Jahan-Parvar, and Hening Liu (2018). Does Smooth Ambiguity Matter for Asset Pricing? International Finance Discussion Papers 1221. https://doi.org/10.17016/IFDP.2018.1221 International Finance Discussion Papers Board of Governors of the Federal Reserve System Number 1221 January 2018

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 1221 January 2018 Does Smooth Ambiguity Matter for Asset Pricing? A. Ronald Gallant, Mohammad R. Jahan-Parvar, and Hening Liu NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at www.ssrn.com.

Does Smooth Ambiguity Matter for Asset Pricing? A. Ronald Gallant Mohammad R. Jahan-Parvar Hening Liu Penn State University∗ Federal Reserve Board† University of Manchester‡§ January 2018 Abstract We use the Bayesian method introduced by Gallant and McCulloch (2009) to estimate consumption-based asset pricing models featuring smooth ambiguity preferences. We rely on semi-nonparametric estimation of a flexible auxiliary model in our structural estimation. Based on the market and aggregate consumption data, our estimation provides statistical support for asset pricing models with smooth ambiguity. Statistical model comparison shows that models with ambiguity, learning and time-varying volatility are preferred to the long-run risk model. We analyze asset pricing implications of the estimated models. JEL Classification: C61; D81; G11; G12 Keywords: Ambiguity, Bayesian estimation, equity premium, Markov-switching, long-run risk ∗Department of Economics, Pennsylvania State University, 511 Kern Graduate Building, University Park, PA 16802 U.S.A. e-mail: aronaldg@gmail.com. †CorrespondingAuthor,BoardofGovernorsoftheFederalReserveSystem,20thSt. NWandConstitutionAve., Washington, DC 20551 U.S.A. e-mail: Mohammad.Jahan-Parvar@frb.gov. ‡Accounting and Finance Group, Alliance Manchester Business School, University of Manchester, Booth Street West, Manchester M15 6PB, UK. e-mail: Hening.Liu@manchester.ac.uk. §We thank Stijn Van Nieuwerburgh (the editor) and two anonymous referees for many helpful comments that substantially improved the paper. We also thank Yoosoon Chang, Stefanos Delikouras, Marco Del Negro, Luca Guerrieri, Philipp K. Illeditsch, Shaowei Ke, Nour Meddahi, Thomas Maurer, Jianjun Miao, James Nason, Joon Y. Park,EricRenault,MichaelStutzer,ToniWhited,seminarparticipantsatFederalReserveBoard,GeorgeWashington University, Georgetown University, Indiana University, North Carolina State University, University of Colorado, UniversityofMaryland,MidwestEconometricGroupmeeting2014,CFE2014,SoFiEannualconference2015,China InternationalConferenceofFinance2015,FMA2015,theEconometricSocietyWorldCongress2015,the2016Winter MeetingoftheEconometricSociety,and2016SFSCavalcadeforhelpfulcommentsanddiscussions. Thispaperwas previously circulated under the title “Measuring Ambiguity Aversion.” The analysis and the conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. Any remaining errors are ours. 1

1 Introduction A number of asset pricing puzzles pose remarkable challenges to the standard consumption-based asset pricing model with a fully rational representative-agent. Among them is the “equity premium puzzle” first documented by Mehra and Prescott (1985), which states that the standard model requires an implausibly high level of risk aversion to explain the historical equity premium in the U.S. data. Other important stylized facts of the stock market include excess return volatility, countercyclical equity premium and equity volatility, and return predictability.1 A recent strand of the literature proposes to embed ambiguity in an otherwise standard model to explain various asset pricing puzzles. An ambiguity-averse agent recognizes uncertainty about an objective law governing the state process and is averse to such uncertainty. There are two popular approaches to model ambiguity in the asset pricing literature, the multiple-priors approach and the smooth ambiguity approach.2 Existing consumption-based models with ambiguity are largely confined to model calibration. However, calibration does not provide the likelihood of the model given observed macroeconomic and financial variables. As a result, statistical support for the importance of ambiguity in asset pricing is still limited in the structural model estimation literature. In this paper, we use the “General Scientific Models” (henceforth, GSM) Bayesian estimation method developed by Gallant and McCulloch (2009) to estimate a set of consumption-based asset pricing models with smooth ambiguity preferences. Our Bayesian estimation jointly produces estimates of preference parameters and parameters governing state dynamics in a structural model. We consider models with an ambiguity-averse representative agent who is uncertain about the conditional mean growth rate of aggregate consumption. The agent’s preferences are represented by generalized recursive smooth ambiguity utility advanced by Hayashi and Miao (2011) and Ju and Miao (2012). This class of preferences builds on the seminal work of Klibanoff, Marinacci, and Mukerji (2005, 2009) and allows for the separation among risk aversion, ambiguity aversion and the elasticity of intertemporal substitution (EIS). Altug, Collard, C¸akmakli, Mukerji, and O¨zs¨oylev 1 See Shiller (1981), Fama and French (1989), Schwert (1989), and Fama and French (1988a) for related empirical evidence. 2 See Epstein and Wang (1994), Chen and Epstein (2002), Epstein and Schneider (2008), and Drechsler (2013) for applications of the multiple-priors preferences and Ju and Miao (2012), and Collard, Mukerji, Sheppard, and Tallon (2017)forapplicationsofsmoothambiguitypreferences. Thesmoothambiguityutilitymodelhasaconnectionwith risk-sensitive control and robustness, see Klibanoff, Marinacci, and Mukerji (2009), Hansen (2007) and Hansen and Sargent (2010). 1

(2017) show that in a special case, where the EIS and relative risk aversion are inversely related, Ju and Miao’s generalized smooth ambiguity utility function is not equivalent to the preferences proposed by Klibanoff et al. (2005, 2009). Our estimation suggests a clear distinction of the EIS from risk aversion for all models and a preference for early resolution of uncertainty. We examine three models featuring smooth ambiguity. The first is the original model of Ju and Miao(2012). ThegrowthrateofconsumptionfollowsaMarkov-switchingprocessinwhichthemean growth rate depends on a hidden state. The hidden state evolves according to a two-state Markov chain. The agent cannot observe the state but can learn about the state in a Bayesian fashion by observing realized growth rates of consumption. Ambiguity arises since the mean growth rate is unobservable. Because the hidden state evolves dynamically over time, learning cannot resolve the agent’s ambiguity in the long run. The agent is ambiguity-averse in that he dislikes a meanpreserving spread in the continuation value led by the agent’s belief about the hidden state. As a result, compared with a solely risk-averse agent, the ambiguity-averse agent effectively assigns more probability weight to “bad” states that are associated with lower levels of the continuation values. Thesecondmodelisanextensionofthefirstandincorporatestime-varyingconditionalvolatility. We postulate that conditional volatility of consumption growth follows another two-state Markov chainthatisindependentofthechainforthemeangrowthstate, asinMcConnellandPerez-Quiros (2000) and Lettau, Ludvigson, and Wachter (2008). A number of studies have examined the role of time-varying volatility and found that volatility risk is significantly priced in the stock market, see Bollerslev, Tauchen, and Zhou (2009), Drechsler (2013) and Bansal, Kiku, Shaliastovich, and Yaron (2014), among others. By estimating a model with ambiguity, learning and time-varying volatility,weaimtoinvestigatewhether(1)inclusionoftime-varyingvolatilityaffectstheestimated impact of ambiguity on asset prices, and (2) the model with time-varying volatility represents an improvement over the original model. The third model is built on the long-run risk model of Bansal and Yaron (2004) and the smooth ambiguity model of Collard, Mukerji, Sheppard, and Tallon (2017). The motivation for examining this model is to study the impact of ambiguity in a long-run risk setting. The persistent long-run risk component in the conditional mean of consumption growth is empirically difficult to detect. 2

Thus, itisreasonabletopostulatethattheagentalsofacesthesamedifficultyasaneconometrician does.3 Similar to the model setup of Ju and Miao (2012), the agent cannot observe the long-run risk component governing mean consumption growth but can learn about it in a Bayesian fashion by observing realizations of consumption and dividend growth rates. In addition, we incorporate stochastic volatility as an exogenous process as in Bansal and Yaron (2004).4 By estimating this model, we want to investigate to what extent the estimated level of ambiguity aversion depends on specifications of state processes and the information structure. In all three models, the agent’s ambiguity aversion endogenously generates pessimistic beliefs about the distribution of consumption growth. In contrast to an ambiguity-neutral investor, the ambiguity-averse agent always slants his belief toward states with low levels of conditional mean growthofconsumption. Thispessimismismanifestedbyasharpincreaseinthepricingkernelwhen the economy experiences a negative shock after staying at the “normal” growth rate for several periods. The pessimistic distortion to the pricing kernel raises its volatility and thus implies a high market price of risk and high equity premium. Inadditiontomodelswithsmoothambiguity,wealsoestimatetwobaselinemodelswithEpstein and Zin (1989)’s recursive utility for model comparison. The first baseline model is the long-run risk model studied by Bansal, Kiku, and Yaron (2012), which is an improved formulation of the original model of Bansal and Yaron (2004). The second baseline model is a special case of Ju and Miao’s model, where we suppress ambiguity aversion. In this model, the agent without endogenous pessimism makes Bayesian inference to evaluate mean consumption growth. By estimating a series of structural models with and without smooth ambiguity, we address two important questions: (1) does a structural estimation with macro-finance data lend statistical support to the class of smooth ambiguity preferences that have sound decision-theoretic basis? (2) Based on a standard Bayesian model comparison between those featuring smooth ambiguity and Epsetin-Zin’s preferences, which estimated model is the preferred model? We find a significant distinction between risk aversion 3 Bidder and Dew-Becker (2016) study a related framework where the agent estimates the consumption process nonparameterically and prices assets using a pessimistic model. They find that long-run risks arise endogenously as the worst-caseoutcome. Collardetal.(2017)consideramoreelaboratemodelinwhichtheagentisnotonlyambiguous aboutthelatentmeangrowthrateofconsumptionbutalsoambiguousaboutwhetherthelatentvariablecomesfrom a highly persistent process or a moderately persistent process. 4 Incorporating an unobservable stochastic volatility component together with learning is beyond the scope of our study. We leave estimation of the model in which the agent also has ambiguity about the volatility state for future research. 3

and ambiguity aversion in the estimated models. Moreover, the distinction is robust to different specifications of consumption dynamics. A model comparison exercise based on posterior likelihoods and the Bayesian information criteria (BIC) shows that the two models featuring smooth ambiguity and time-varying volatility are preferredtoboththelong-runriskmodelandtheEpstein-Zin’srecursiveutilitymodelwithregimeswitching consumption growth and learning. Prior to our study, Bansal, Gallant, and Tauchen (2007) and Aldrich and Gallant (2011) concluded that the long-run risk model is a preferred model. In addition, we find that the estimated smooth ambiguity models can match moments of asset returns better than the Epstein-Zin’s models do. The estimated Ju and Miao’s model, which receives less statistical support than the long-run risk model, matches the equity premium and variance risk premium in the data well. We use the projection method to solve all models examined in this paper. The log-linear approximation method that has been widely used in the long-run risk literature is not applicable to our smooth ambiguity models. This is because learning induces nonlinearities in the dynamics of the agent’s beliefs. Additionally, the smooth ambiguity utility function is highly nonlinear. To keep our quantitative analysis consistent, we also use the projection method to solve the long-run risk model. In a recent work, Pohl, Schmedders, and Wilms (2017) assess numerical accuracy of the log-linear approximation method and find that applying log-linearization to solve long-run risk models can yield biased results due to neglecting higher-order effects. The bias becomes more pronounced when the long-run risk and stochastic volatility components are highly persistent. Using the log-linear approximation and a mixed data frequency approach, Schorfheide, Song, and Yaron (2017) perform Bayesian estimation of long-run risk models with several specifications of stochastic volatility and find that the long-run risk component and stochastic volatilities are highly persistent. While our estimation is based on annual data and Bayesian indirect inference, we also find persistent long-run risk and stochastic volatility components as well as a high EIS. Similar to other macro-finance applications, we face sparsity of data because we use annual data for estimation. In addition, the likelihood of a structural asset pricing model is not readily available. As has become standard in the macro-finance empirical literature, we use prior information and a Bayesian estimation methodology to overcome data sparsity. Specifically, we use 4

the GSM Bayesian estimation method developed by Gallant and McCulloch (2009). GSM is the Bayesian counterpart to the classical “indirect inference” and “efficient method of moments” (hereafter, EMM) methods introduced by Gouri´eroux, Monfort, and Renault (1993) and Gallant and Tauchen (1996, 1998, 2010). These are simulation-based inference methods that rely on an auxiliary model for implementation. The Bayesian estimation of GSM relies on the theoretical results of Gallant and Long (1997) in its construction of a likelihood. In particular, Gallant and McCulloch synthesize a likelihood by means of an auxiliary model and simulations from the structural model. A comparison of Aldrich and Gallant (2011) with Bansal et al. (2007) displays the advantages of a Bayesian EMM approach relative to a frequentist EMM approach, particularly for the purpose of model comparison. An indirect inference approach is an appropriate estimation methodology in the context of this study since the estimated equilibrium model is highly nonlinear and does not admit analytically tractable solutions, thereby severely inhibiting accurate numerical construction of a likelihood by means other than GSM. GSM uses a sieve (see Section 3) specially tailored to macroeconomic and financial time-series applications as the auxiliary model. When a suitable sieve is used as the auxiliary model, as in this study, the GSM method synthesizes the exact likelihood implied by the model.5 In this instance, the synthesized likelihood model departs significantly from a normal-errors likelihood, which suggests that alternative econometric methods based on normal approximations will give biased results. In particular, in addition to the generalized autoregressive conditional heteroscedasticity (GARCH) effect, the four-dimensional error distribution implied by the smooth ambiguity model is skewed in all four components and has fat-tails for consumption growth, dividend growth and stock returns, and thin tails for bond returns. Thispapercontributestoagrowingbodyofliteratureonambiguity,learningandmacro-finance. Wediscusscloselyrelatedpapershere. EpsteinandSchneider(2007)developamodelwithlearning under ambiguity. They use the multiple-priors approach to model ambiguity and assume a set of priors and a set of likelihoods for signals. Beliefs are updated by Bayes’ rule in an appropriate way. Epstein and Schneider (2008) apply this model to study information quality and asset prices. Leippold, Trojani, and Vanini (2008) adopt the continuous-time multiple-priors framework of Chen 5 GallantandMcCulloch(2009)usetheterms“scientificmodel”and“statisticalmodel”insteadoftheterms“structural model” and “auxiliary model” used in the indirect inference econometric literature. We will follow the conventions of the econometric literature. The structural models here are equilibrium asset pricing models. 5

and Epstein (2002) to analyze asset pricing implications of learning under ambiguity. Cogley and Sargent (2008) examine the impacts of pessimistic beliefs on the market price of risk and equity premium. Hansen and Sargent (2010) consider robustness concerns in learning and study timevarying model uncertainty premia. Collard et al. (2017) assume that a representative agent with smooth ambiguity preferences faces both model uncertainty and state uncertainty and analyze the dynamics of risk premia conditioning on the historical data. Johannes, Lochstoer, and Mou (2016) and Collin-Dufresne, Johannes, and Lochstoer (2016) study parameter learning and asset prices in the consumption-based framework with recursive utility. Jeong, Kim, and Park (2015) estimate an asset pricing model in which the agent has multiple-priors utility. Their estimation results suggest that ambiguity on the true probability law governing fundamentals carries a sizable premium. Jahan-Parvar and Liu (2014), Backus, Ferriere, and Zin (2015) and Altug et al. (2017) examine both business cycle and asset pricing implications in dynamic stochastic general equilibrium (DSGE) models with smooth ambiguity. Ilut and Schneider (2014) estimate a DSGE model with multiple-priorsutility. Theirestimationsuggeststhattime-varyingconfidenceinfuturetotalfactor productivity explains a significant fraction of the business cycle fluctuations. Bianchi, Ilut, and Schneider (2016) estimate another DSGE model to explain joint dynamics of asset prices and real economic activity in the postwar data. They show that time-varying ambiguity about corporate profits leads to high equity premium and excess volatility. They further show that the recursive multiple priors utility model provides a tractable way to analyze DSGE models with time varying uncertainty and facilitates estimation by means of likelihood methods. The rest of the paper proceeds as follows. Section 2 presents consumption-based asset pricing models with smooth ambiguity. Section 3 discusses the estimation method and empirical findings. Section 4 presents asset pricing implications. Section 5 concludes. Numerical solution methods and additional results are included in the Internet Appendix. 2 Asset Pricing Models The intuitive notions behind any consumption-based asset pricing model are that agents receive income (wage, interest, and dividends) that they use to purchase consumption goods. Agents reallocate their consumption over time by trading stocks that pay random dividends and bonds 6

that pay interest with certainty. This is done for consumption smoothing over time. The budget constraint implies that the purchase of consumption, bonds, and stocks cannot exceed income in any period. Agents are endowed with a utility function that depends on the entire consumption path. The first-order conditions of their utility maximization deliver an intertemporal relation of prices of stocks and bonds. Among all tradable assets, we focus on the risky asset that pays aggregate dividends and the one-period risk-free bond with zero net supply. 2.1 Asset Pricing Models Featuring Smooth Ambiguity We examine three consumption-based asset pricing models in which a representative agent is endowed with smooth ambiguity preferences. These models include (1) Ju and Miao (2012)’s model in which the mean of consumption growth follows a hidden Markov chain with two states, abbreviated as “AAMS”, (2) an extended version of Ju and Miao’s model with time-varying conditional volatility, abbreviated as “AAMSSV”, and (3) a long-run risk model featuring ambiguity in which the long-run risk component is assumed to be unobservable, abbreviated as “AALRRSV” model. The latter model shares many features with the models introduced by Collard et al. (2017). In all these models, the agent cannot observe the state determining mean consumption growth but learns about the state in a Bayesian fashion. The unobservable mean growth state implies that the agent is ambiguous about the data-generating process of fundamentals. Smooth ambiguity utility captures the agent’s aversion toward this ambiguity. 2.1.1 The AAMS model Aggregate consumption follows the process (cid:18) (cid:19) C t ∆c ≡ ln = µ(s )+σ (cid:15) , (cid:15) ∼ N (0,1), t t c c,t c,t C t−1 where (cid:15) is an i.i.d. standard normal random variable, and s indicates the state of mean conc,t t sumption growth and follows a two-state Markov chain. Suppose that “l” and “h” indicate low and high mean growth states respectively. The transition probabilities are given by Pr(s = l|s = l) = p , Pr(s = h|s = h) = p t t−1 ll t t−1 hh 7

Because aggregate dividends are more volatile than aggregate consumption (see Abel, 1999 and Bansal and Yaron, 2004), the dividend growth process is given by (cid:18) (cid:19) D t ∆d ≡ ln = λ∆c +g +σ˜ (cid:15) (1) t t d d d,t D t−1 where (cid:15) is an i.i.d. standard normal random variable that is independent of all other shocks d,t in the model. The parameter λ represents the leverage ratio; see Abel (1999). We pin down the parameters g and σ˜ by the estimates of unconditional mean and volatility of dividend growth. d d We set the unconditional mean of dividend growth to that of consumption growth implied by the Markov-switching model. In addition, we denote the unconditional volatility of dividend growth by σ . d The agent cannot observe the mean growth state but can learn about it through observing the history of consumption and dividends. The agent knows the parameters in the consumption and dividend processes, namely, {µ ,µ ,p ,p ,σ ,λ,g ,σ˜ }. Suppose that the agent’s belief is l h ll hh c d d π = Pr(s = h|I )whereI denotesinformationavailableattimet. Withrespecttolearningabout t t t t the unobservable state, dividends do not contain additional information compared to consumption. As a result, given the prior belief π and full information, the agent updates his beliefs according 0 to Bayes’ rule: p f(∆c |s = h)π +(1−p )f(∆c |s = l)(1−π ) hh t+1 t+1 t ll t+1 t+1 t π = t+1 f(∆c |s = h)π +f(∆c |s = l)(1−π ) t+1 t+1 t t+1 t+1 t where f(∆c |s ) is conditional density with mean µ(s ) and variance σ2: t t t c (cid:34) (cid:35) (∆c −µ(s ))2 t t f(∆c |s ) ∝ exp − . t t 2σ2 c The generalized recursive smooth ambiguity utility function proposed by Hayashi and Miao (2011) and Ju and Miao (2012) implies that given consumption plans C = (C ) the value t t≥0 function V = V (C;π ) is given by t t (cid:104) (cid:105) 1 V (C;π ) = (1−β)C 1−1/ψ +β{R (V (C;π ))}1−1/ψ 1−1/ψ , t t t t t+1 t+1 8

where β ∈ (0,1) is the subjective discount factor, ψ is the elasticity of intertemporal substitution (EIS)parameter,γ isthecoefficientofrelativeriskaversion,andR (V (C ;π ))isthecertainty t t+1 t+1 equivalent of the continuation value given by (cid:18) (cid:20) (cid:16) (cid:104) (cid:105)(cid:17)1−η(cid:21)(cid:19) 1− 1 η R (V (C;π )) = E E V (C;π )1−γ 1−γ . (2) t t+1 t+1 πt {st+1,,t} t+1 t+1 Ambiguity aversion is characterized by the parametric restriction η > γ, where η is the ambiguity aversion parameter. By setting η = γ, we obtain Epstein-Zin’s recursive utility under ambiguity neutrality.6 In the certainty equivalent (2), the expectation operator E [·] is taken with respect st+1,t totheconditionaldistributionofconsumptiongrowthinstates andallotherinformationattime t+1 t. TheexpectationoperatorE istakenwithrespecttotheposteriorbeliefabouttheunobservable πt state. Following Hayashi and Miao (2011), the stochastic discount factor (SDF) in this model is given by (cid:16) (cid:104) (cid:105)(cid:17) 1 −(η−γ). M = β (cid:18) C t+1 (cid:19)−1/ψ(cid:18) V t+1 (cid:19)1/ψ−γ  E {st+1,t} V t 1 + − 1 γ 1−γ  t,t+1 C R (V )  R (V )  t t t+1 t t+1 The last multiplicative term in the SDF arises due to ambiguity aversion. This term makes the SDF more countercyclical than in the case of Epstein and Zin’s recursive utility and induces large variations in the SDF. The risk-free rate, Rf, is the reciprocal of the conditional expectation of the t SDF, 1 Rf = . t E [M ] t t,t+1 P +D t+1 t+1 Stock returns, defined by R = , satisfy the Euler equation t+1 P t E [M R ] = 1. t t,t+1 t+1 6 WefollowJuandMiao(2012)anddonotconsiderη<γ inourestimationasthisparametricrestrictionmightimply “ambiguity loving”, see also Hayashi and Miao (2011). 9

We rewrite the Euler equation as (cid:104) (cid:16) (cid:17)(cid:105) (cid:104) (cid:16) (cid:17)(cid:105) 0 = π˜ E MEZ R −Rf +(1−π˜ )E MEZ R −Rf , t h,t t,t+1 t+1 t t l,t t,t+1 t+1 t where E [·] denotes E [·] for s = h and similarly for state l. We interpret the term MEZ h,t st+1,t t+1 t,t+1 as the SDF under recursive utility: MEZ = β (cid:18) C t+1 (cid:19)− ψ 1 (cid:18) V t+1 (cid:19) ψ 1−γ . zt+1,t+1 C R (V ) t t t+1 We interpret π˜ as the ambiguity-distorted belief and represent it by: t (cid:16) (cid:104) (cid:105)(cid:17)−η−γ π E V1−γ 1−γ t h,t t+1 π˜ = . t (cid:16) (cid:104) (cid:105)(cid:17)−η−γ (cid:16) (cid:104) (cid:105)(cid:17)−η−γ π E V1−γ 1−γ +(1−π ) E V1−γ 1−γ t h,t t+1 t l,t t+1 As long as η > γ, distorted beliefs are not equivalent to Bayesian beliefs. The distortion driven by ambiguity aversion is an equilibrium outcome and implies pessimistic beliefs; see Section 4. We rewrite the Euler equation to solve for the price-dividend ratio, (cid:20) (cid:18) (cid:19) (cid:21) P P D t = E M 1+ t+1 t+1 . t t,t+1 D D D t t+1 t Since Pt is a functional of the state variable π , Pt = Φ(π ), the Euler equation becomes Dt t Dt t Φ(π ) = E [M (1+Φ(π ))exp(∆d )]. t t t,t+1 t+1 t+1 2.1.2 The AAMSSV model We follow McConnell and Perez-Quiros (2000) and Lettau et al. (2008) and extend Ju and Miao’s model by incorporating time-varying conditional volatility. We assume that the conditional mean and volatility states follow two independent Markov chains. The consumption process takes the form ∆c = µ(sµ)+σ(sσ)(cid:15) , (cid:15) ∼ N (0,1) t t t c,t c,t 10

with transition probabilities Pr (cid:0) sµ = l|sµ = l (cid:1) = pµ, Pr (cid:0) sµ = h|sµ = h (cid:1) = pµ , t t−1 ll t t−1 hh Pr (cid:0) sσ = l|sσ = l (cid:1) = pσ, Pr (cid:0) sσ = h|sσ = h (cid:1) = pσ . t t−1 ll t t−1 hh To ease the analysis, we assume that the mean state sµ is unobservable while the volatility state t sσ is observable. We make this simplifying assumption for two reasons: First, empirical studies t such as Bryzgalova and Julliard (2015) have established that estimation and characterization of the mean consumption growth is more difficult than consumption volatility. Second, according to the existing literature, volatility states are very persistent, leading to filtered probabilities of the volatility state close to 1. These results imply that ambiguity has little room with respect to consumption volatility states.7 The agent updates beliefs according to Bayes’ rule as pµ f (cid:0) ∆c |sµ = h,sσ (cid:1) π + (cid:0) 1−pµ(cid:1) f (cid:0) ∆c |sµ = l,sσ (cid:1) (1−π ) π = hh t+1 t+1 t+1 t ll t+1 t+1 t+1 t t+1 f (cid:0) ∆c |sµ = h,sσ (cid:1) π +f (cid:0) ∆c |sµ = l,sσ (cid:1) (1−π ) t+1 t+1 t+1 t t+1 t+1 t+1 t where f (cid:0) ∆c |sµ ,sσ (cid:1) is conditional density t+1 t+1 t+1 f (cid:0) ∆c |sµ ,sσ (cid:1) ∝ 1 exp (cid:34) − (cid:0) ∆c t+1 −µ (cid:0) sµ t+1 (cid:1)(cid:1)2(cid:35) t+1 t+1 t+1 σ (cid:0) sσ (cid:1) 2σ (cid:0) sσ (cid:1)2 t+1 t+1 The value function is given by (cid:104) (cid:105) 1 V (C;π ,sσ) = (1−β)C 1−1/ψ +β (cid:8) R (cid:0) V (cid:0) C;π ,sσ (cid:1)(cid:1)(cid:9)1−1/ψ 1−1/ψ , t t t t t t+1 t+1 t+1 R (cid:0) V (cid:0) C;π ,sσ (cid:1)(cid:1) = (cid:18) E (cid:20) (cid:16) E (cid:104) V (cid:0) C;π ,sσ (cid:1)1−γ (cid:105)(cid:17) 1 1 − − γ η(cid:21)(cid:19) 1− 1 η t t+1 t+1 t+1 πt {sµ ,sσ,t} t+1 t+1 t+1 t+1 t in which E [·] denotes the expectation conditional on the history up to time t including {sµ ,sσ,t} t+1 t the volatility state sσ, and a probability distribution of consumption growth given state sµ . The t t+1 7 We have also examined the model in which both the conditional mean and volatility states are unobservable. But solving the model requires substantial run time to achieve convergence. For some parameter values, the numerical algorithmfailstolocateafixedpointforthewealth-consumptionratio. ThesedifficultiesmakeourBayesianMCMC estimationinfeasible. Lettauetal.(2008)alsopointouttheconvergenceissueforEpsteinandZin’srecursiveutility. 11

conditional expectation can be explicitly written as  (cid:104) (cid:105) (cid:104) (cid:105) E (cid:104) V1−γ (cid:105) =   pσ ll E {sµ t+1 ,sσ t+1 ,t} V t 1 + − 1 γ|sσ t+1 = l +(1−pσ ll )E {sµ t+1 ,sσ t+1 ,t} V t 1 + − 1 γ|sσ t+1 = h ,sσ t = l {sµ ,sσ,t} t+1 (cid:104) (cid:105) (cid:104) (cid:105) t+1 t   (1−pσ hh )E {sµ ,sσ ,t} V t 1 + − 1 γ|sσ t+1 = l +pσ hh E {sµ ,sσ ,t} V t 1 + − 1 γ|sσ t+1 = h ,sσ t = h t+1 t+1 t+1 t+1 where (cid:104) (cid:105) (cid:90) 1 (cid:32) (cid:0) ∆c −µ (cid:0) sµ (cid:1)(cid:1)2(cid:33) E V1−γ ∝ exp − t+1 t+1 V1−γd(∆c ). {sµ t+1 ,sσ t+1 ,t} t+1 σ (cid:0) sσ (cid:1) 2σ (cid:0) sσ (cid:1)2 t+1 t+1 t+1 t+1 The SDF in this model is (cid:16) (cid:104) (cid:105)(cid:17) 1 −(η−γ) M = β (cid:18) C t+1 (cid:19)−1/ψ(cid:18) V t+1 (cid:19)1/ψ−γ  E {sµ t+1 ,sσ t ,t} V t 1 + − 1 γ 1−γ  t,t+1 C R (V )  R (V )  t t t+1 t t+1 The dividend growth process is specified in the same form as in the AAMS model, i.e., in equation (1). Stockreturnsandtherisk-freeratearedefinedinasimilarwayaccordingly. Theprice-dividend ratio (Pt = Φ(π ,sσ)) satisfies the Euler equation Dt t t Φ(π ,sσ) = E (cid:2) M (cid:0) 1+Φ (cid:0) π ,sσ (cid:1)(cid:1) exp(∆d ) (cid:3) t t t t,t+1 t+1 t+1 t+1 2.1.3 The AALRRSV model We consider the long-run risk model of Bansal and Yaron (2004), the specification of which is given by ∆c = µ +x +σ (cid:15) t+1 c t+1 t c,t+1 ∆d = µ +λx +ϕ σ (cid:15) t+1 d t+1 d t d,t+1 x = ρ x +ϕ σ (cid:15) t+1 x t x t x,t+1 σ2 = µ2 +ρ (cid:0) σ2−µ2(cid:1) +σ (cid:15) t+1 σ s t s w w,t+1 (cid:15) ,(cid:15) ,(cid:15) ,(cid:15) ∼ i.i.d.N (0,1). c,t+1 d,t+1 x,t+1 w,t+1 12

In Bansal and Yaron’s calibration, x is a highly persistent component, and σ is the stochastic t t volatility component representing time-varying economic uncertainty that is also highly persistent. Thelong-runrisksliteratureassumesthatx isfullyobservableandthusappearsasastatevariable t in the wealth-consumption ratio and price-dividend ratio. However, this component is difficult to identify using empirically observed economic variables as documented by Bansal et al. (2007), Ma (2013), and Johannes et al. (2016), among others. The difficulty in estimating x gives rise to the t agent’s ambiguity about the mean consumption growth. As a result, we adopt a more plausible information structure by assuming that x is unobservable. Collard et al. (2017) provide ample t theoretical support for this assumption. In particular, we maintain that the agent observes the realizations of ∆c and ∆d contemt+1 t+1 poraneously but never observes the realization of x or ((cid:15) ,(cid:15) ,(cid:15) ). This feature of the model t c,t d,t x,t characterizes ambiguity, i.e., the agent’s lack of confidence in estimating the conditional mean of consumption growth. Instead, the agent uses consumption and dividend growth realizations to filter the unobserved long-run risk component x . To make the model tractable and comparable t to the long-run risks model, we assume that the conditional volatility of consumption growth σ is t observable. We also assume that values of the parameter vector (µ ,µ ,ϕ ,ϕ ,ϕ ,ρ ,λ,µ ,ρ ,σ ) c d c d x x s s w are known to the agent. Supposethatx hasaGaussiandistribution. ThestandardKalmanfilterimpliesthattheagent 0 updatesbeliefsaccordingtoBayes’ruleconditionalonthehistoryofrealizationsof∆c and∆d t+1 t+1 given the Gaussian prior. The updated belief is also Gaussian with mean xˆ and variance ν , t+1 t+1 (cid:104) (cid:105) (cid:0) (cid:1)2 i.e., x ∼ N (xˆ ,ν ). We define xˆ = E[x |I ] and ν = E x −xˆ |I . It t+1 t+1 t+1 t+1|t t+1 t t+1|t t+1 t+1|t t immediately follows that xˆ = ρ xˆ , and ν = ρ2ν +ϕ2σ2. t+1|t x t t+1|t x t x t 13

The Kalman filter implies the following updating equations   (cid:20) (cid:21) vc xˆ t+1 = xˆ t+1|t +ν t+1|t 1 λ F t − + 1 1|t   t+1|t   vd t+1|t (cid:20) (cid:21) (cid:20) (cid:21)(cid:48) ν t+1 = ν t+1|t −ν t 2 +1|t 1 λ F t − + 1 1|t 1 λ where F is given by t+1|t   ν +σ2 λν F =  t+1|t t t+1|t  t+1|t   λν λ2ν +ϕ2σ2 t+1|t t+1|t d t (cid:20) (cid:21) and the innovation vector vc vd is given by t+1|t t+1|t     vc ∆c −µ −ρ xˆ  t+1|t  =  t+1 c x t .     vd ∆d −µ −λρ xˆ t+1|t t+1 d x t This model has three state variables (xˆ ,ν ,σ ). The value function under smooth ambiguity t t t utility V = V (C;xˆ ,ν ,σ ) satisfies t t t t t (cid:104) (cid:105) 1 V = (1−β)C 1−1/ψ +β{R (V )}1−1/ψ 1−1/ψ , t t t t+1 (cid:18) (cid:20) (cid:16) (cid:104) (cid:105)(cid:17)1−η(cid:21)(cid:19) 1− 1 η R (V ) = E E V1−γ 1−γ . t t+1 {xˆt,νt} {xt,σt,t} t+1 The certainty equivalent R (V ) reflects the agent’s aversion toward ambiguity in estimating the t t+1 long-run risk component x . The agent lacks confidence in the Gaussian posterior of x and thus t t appliespessimisticdistortiontotheposterior. ThisdistortionisvisibleinFigure1. Inwhatfollows, we describe the mechanism of how ambiguity aversion leads to distortion in the posterior. The SDF in this model is (cid:16) (cid:104) (cid:105)(cid:17) 1 −(η−γ) M = β (cid:18) C t+1 (cid:19)− ψ 1 (cid:18) V t+1 (cid:19) ψ 1−γ  E {xt,σt,t} V t 1 + − 1 γ 1−γ  . t,t+1 C R (V )  R (V )  t t t+1 t t+1 14

We solve the price-dividend ratio, Pt = Φ(xˆ ,ν ,σ ), from the Euler equation Dt t t t Φ(xˆ ,ν ,σ ) = E [M (1+Φ(xˆ ,ν ,σ ))exp(∆d )]. t t t t t,t+1 t+1 t+1 t+1 t+1 Given the Gaussian posterior obtained according to Bayes’ rule, x ∼ N (xˆ ,ν ), we derive the t t t distorted density of x due to ambiguity aversion. The SDF M can be decomposed as M = t t,t+1 t,t+1 MEZ MAA in which MEZ and MAA are given respectively by t,t+1 t t,t+1 t (cid:16) (cid:104) (cid:105)(cid:17) 1 −(η−γ) MEZ = β (cid:18) C t+1 (cid:19)− ψ 1 (cid:18) V t+1 (cid:19) ψ 1−γ ,MAA =  E {xt,σt,t} V t 1 + − 1 γ 1−γ  . t,t+1 C R (V ) t  R (V )  t t t+1 t t+1 The Euler equation can be rewritten as  (cid:16) (cid:104) (cid:105)(cid:17) 1 −(η−γ) E V1−γ 1−γ 0 = E t    M t E ,t Z +1 (cid:16) R t+1 −R t f (cid:17)   {xt,σ R t,t t } (V t+ t+ 1 1 )      . By the law of iterated expectations, we obtain (cid:16) (cid:104) (cid:105)(cid:17)−η−γ 0 = (cid:90) E (cid:104) MEZ (cid:16) R −Rf (cid:17) |x (cid:105) E t V t 1 + − 1 γ|x t 1−γ f(x t |xˆ t ,ν t ) dx (3) t t,t+1 t+1 t t (cid:82) (cid:16) E (cid:104) V1−γ|x (cid:105)(cid:17)−η 1− − γ γ f(x |xˆ ,ν )dx t t t+1 t t t t t where f(x |xˆ ,ν ) denotes the Bayesian density of x given xˆ and ν . It is clear from (3) that the t t t t t t distorted density driven by ambiguity, f˜(x |xˆ ,ν ,t), is given by t t t (cid:16) (cid:104) (cid:105)(cid:17)−η−γ E V1−γ|x 1−γ f˜(x |xˆ ,ν ,t) = t t+1 t f(x |xˆ ,ν ) t t t (cid:82) (cid:16) E (cid:104) V1−γ|x (cid:105)(cid:17)−η 1− − γ γ f(x |xˆ ,ν )dx t t t t t+1 t t t t t 15

2.2 Alternative Models Featuring Ambiguity Neutral Preferences The recursive utility function of Epstein and Zin (1989) takes the form (cid:20) (cid:21) 1 V (C) = (1−β)C 1−1/ψ +β (cid:110) E (cid:16) V (C)1−γ (cid:17)(cid:111)1−1/ψ 1−1/ψ , t t t t+1 As usual, the SDF under recursive utility, denoted by MEZ, is t+1 MEZ = β (cid:18) C t+1 (cid:19)− ψ 1 (cid:18) V t+1 (cid:19) ψ 1−γ . (4) t,t+1 C E (V ) t t t+1 By setting η = γ in the generalized recursive smooth ambiguity utility function, we suppress ambiguity aversion and obtain Epstein-Zin’s recursive utility model as a special case. We impose this parametric restriction to obtain model “EZMS” as the ambiguity-neutral version of model AAMS. The second alternative model is the long-run risk model of Bansal et al. (2012), which we label as “EZLRRSV”. The model specification is ∆c = µ +x +σ (cid:15) t+1 c t t c,t+1 ∆d = µ +λx +ϕ σ (cid:15) +ϕ σ (cid:15) t+1 d t+1 d t d,t+1 c t c,t+1 x = ρ x +ϕ σ (cid:15) t+1 x t x t x,t+1 σ2 = µ2 +ρ (cid:0) σ2−µ2(cid:1) +σ (cid:15) t+1 σ σ t σ w w,t+1 (cid:15) ,(cid:15) ,(cid:15) ,(cid:15) ∼ i.i.d.N (0,1). c,t+1 d,t+1 x,t+1 w,t+1 with notations defined in the same way as in model AALRRSV. The two state variables are x and t σ2. The price-dividend ratio, Pt = Φ (cid:0) x ,σ2(cid:1) , satisfies the Euler equation t Dt t t Φ (cid:0) x ,σ2(cid:1) = E (cid:2) M (cid:0) 1+Φ (cid:0) x ,σ2(cid:1)(cid:1) exp(∆d ) (cid:3) . t t t t,t+1 t t t+1 We present the structural parameters to be estimated for each model in Table 2. In estimating models AALRRSV and EZLRRSV, we impose that µ = µ . We solve all the models examined in c d this paper using the collocation projection method with Chebyshev polynomials. Pohl et al. (2017) 16

show that this is a reliable solution method for nonlinear asset pricing models. The details of the implementation and numerical accuracy assessment are available in the Internet Appendix. 3 Data and the Estimation Method 3.1 Data Throughout this paper, lower case denotes the natural logarithm of an upper case variable; e.g., c = ln(C ), where C is the observed consumption in period t, and d = ln(D ), where D is t t t t t t dividends paid in period t. Similarly, we use logarithmic risk-free interest rate (rf) and aggregate t equity market return inclusive of dividends (r = ln(P +D )−lnP ) in the analysis, where P t t t t−1 t is the stock price in period t. We use real annual data from 1941 to 2015. The sample period 1941–1949 provides initial lags for the recursive parts of our estimation and the sample period 1950–2015 yields estimation results and diagnostics. Our measure for the risk-free rate is the one-year U.S. Treasury Bill rate. To construct the real risk-free rate, we regress the ex-post real one-year Treasury Bill yield on the nominal rate and past annual inflation, available from Wharton Research Data Services (WRDS) TreasuryandInflationdatabase. Thefittedvaluesfromthisregressionaretheproxyfortheex-ante real interest rate. Using other estimates of expected inflation to construct the real rate does not lead to significant changes in our results. Our proxy for risky assets is the value-weighted returns (including dividends) on the aggregate stock market portfolio of the NYSE/AMEX/NASDAQ, which is obtained from the Center for Research in Security Prices (CRSP) and deflated using the CPI data. We use the sum of real nondurable and services consumption, items 16 and 17 on the NIPA Table 7.1 “Selected Per Capita Product and Income Series in Current and Chained Dollars,” published by the Bureau of Economic Analysis (BEA) as our measure of real consumption. These values are reported in chained 2009 U.S. Dollars and constructed using mid-year population data. We construct the dividend growth rate series by first computing the gross dividend level from the value-weighted returns including and excluding dividends and lagged index levels. We then obtain the real dividend growth rate by deflating the nominal growth rate. Table 1 presents the summary statistics of the data used in estimation. The p-values of Jarque 17

and Bera (1980) test of normality imply that the assumption of normality is not rejected for the consumption growth series, but it is rejected for all other variables. Real equity returns, interest rates, and dividend growth rates all exhibit negative skewness. In addition, both real interest rates and dividend growth rates show significant excess kurtosis. Figure 2 plots the data. 3.2 GSM: Estimation of the structural model To estimate model parameters we use a Bayesian method proposed by Gallant and McCulloch (2009), abbreviated GM hereafter, which they termed General Scientific Models (GSM). The GSM methodologywasrefinedinAldrichandGallant(2011), abbreviatedAGhereafter.8 Thediscussion here incorporates those refinements and is to a considerable extent a paraphrase of AG. GSM is a Bayesian counterpart of the indirect inference method of Gouri´eroux, Monfort, and Renault(1993)andtheefficientmethodofmomentsofGallantandTauchen(1996,1998). Assuch, implementingthisestimationmethodrequiresfittingthedatawithanover-parameterizedauxiliary model (not rooted in theory) and then recovering parameter estimates from the structural model (founded on theory) by computing the mapping linking the parameter spaces of these two models. We discuss the estimation method pertaining to the structural model and the map in detail, and then discuss the auxiliary model and its estimation briefly. Let the transition density of a structural model be denoted by p(y |z ,θ), θ ∈ Θ, t t−1 where y is the vector of observable variables, z = (y ,...,y ) if Markovian and z = t t−1 t−1 t−L t−1 (y ,...,y )ifnot,andΘisthestructuralparameterspace. Asaresult,z servesasashorthand t−1 1 t−1 for lag-lengths that are generally greater than 1. Thus, transition densities may depend on L-lags of the data (if Markovian) or the entire history of observations (if non-Markovian). There are five structural models under consideration in this application: the three models featuring smooth ambiguity and the two alternative models with Epstein-Zin’s recursive utility, all of which are Markovian and described in Section 2. 8 The code implementing the method with AG refinements, together with a User’s Guide, is in the public domain at http://www.aronaldg.org/webfiles/gsm. 18

Wepresumethatthereisnostraightforwardalgorithmforcomputingthelikelihoodbutthatwe can simulate data from p(·|·,θ) for a given θ ∈ Θ. We presume that simulations from the structural model are ergodic. We assume that there is a transition density f (called the auxiliary model) f(y |z ,ω), ω ∈ Ω t t−1 and Ω is the auxiliary model parameter space. In addition, we assume that a map exists g : θ (cid:55)→ ω such that p(y |z ,θ) = f(y |z ,g(θ)), θ ∈ Θ. (5) t t−1 t t−1 We assume that f(y |z ,ω) and its gradient (∂/∂ω)f(y |z ,ω) are fairly easy to evaluate. Then t t−1 t t−1 g is called the “implied map”.9 When Equation (5) holds, f is said to “nest” p. Whenever we need the likelihood (cid:81)n p(y |z ,θ), we use t=1 t t−1 n (cid:89) L(θ) = f(y |z ,g(θ)), (6) t t−1 t=1 where{y ,z }n arethedataandnisthesamplesize. AftersubstitutingL(θ)for (cid:81)n p(y |z ,θ), t t−1 t=1 t=1 t t−1 standard Bayesian MCMC methods become applicable. That is, we have a likelihood L(θ) from Equation (6) and a prior ξ(θ) from Subsection 3.5 that are sufficient for us to implement Bayesian methods by means of MCMC. A good introduction to these methods is Gamerman and Lopes (2006). The difficulty in implementing GM’s proposal is to compute the implied map g accurately enough that the accept/reject decision in an MCMC chain (Step 5 in the algorithm below) is correct when f is a nonlinear model. The algorithm proposed by AG to address this difficulty is described next. 9 Gouri´eroux, Monfort, and Renault (1993), Gallant and Tauchen (1996), Gallant and McCulloch (2009), and Gallant and Tauchen (2010) provide rigorous support for conditions ensuring that the auxiliary model f is a good approximation for the structural model p. 19

Given θ, ω = g(θ) is computed by minimizing Kullback-Leibler divergence (cid:90) (cid:90) d(f,p) = [logp(y|z,θ)−logf(y|z,ω)] p(y|z,θ)dyp(z|θ)dz with respect to ω. The advantage of Kullback-Leibler divergence over other distance measures is (cid:82)(cid:82) that the part that depends on the unknown p(·|·,θ), logp(y|z,θ)p(y|z,θ)dyp(z|θ)dz, does not have to be computed to solve the minimization problem. We approximate the integral that must be computed by (cid:90) (cid:90) N 1 (cid:88) logf(y|z,ω)p(y|z,θ)dyp(z|θ)dx ≈ logf(yˆ|zˆ ,ω), t t−1 N t=1 where {yˆ,zˆ }N is a simulation of length N from p(·|·,θ). Upon dropping the division by N, t t−1 t=1 the implied map is computed as N (cid:88) g : θ (cid:55)→ argmax logf(yˆ |zˆ ,ω). (7) t t−1 ω t=1 We use N = 1000 in the estimation of all the five models. Results (posterior means, posterior standard deviations, etc.) are not sensitive to N; doubling N makes no difference other than doubling computational time. It is essential that the same seed be used to start these simulations so that the same θ always produces the same simulation. GMrunaMarkovchain{ω }K oflengthK tocomputeωˆ thatsolvesexpression(7). Thereare t t=1 two other Markov chains discussed below and so this chain is called the ω-subchain to distinguish among them. While the ω-subchain must be run to provide the scaling for the model assessment method that GM propose, the ωˆ that corresponds to the maximum of (cid:80)N logf(yˆ |zˆ ,ω) over t=1 t t−1 the ω-subchain is not a sufficiently accurate evaluation of g(θ) for our auxiliary model. This is mainlybecauseourauxiliarymodelisamultivariateGARCHspecificationofBollerslev(1986)that Engle and Kroner (1995) call BEKK. Likelihoods incorporating BEKK are notoriously difficult to optimize. AGuseωˆ asastartingvalueandmaximizetheexpression(7)usingtheBFGSalgorithm, see Fletcher (1987). This also is not a sufficiently accurate evaluation of g(θ). A second refinement is necessary. The second refinement is embedded within the MCMC chain {θ }H of length H t t−1 20

that is used to compute the posterior distribution of θ. It is called the θ-chain. The θ-chain is generated using the Metropolis algorithm. The Metropolis algorithm is an iterative scheme that generates a Markov chain whose stationary distribution is the posterior of θ. To implement it, we require a likelihood, a prior, and transition density in θ called the proposal density. The likelihood is Equation (6) and the prior, ξ(θ), is described in Section 3.5. The prior may require quantities computed from the simulation {yˆ,zˆ }N that are used in t t−1 t−1 computing Equation (6). In particular, quantities computed in this fashion can be viewed as the evaluation of a functional of the structural model of the form p(·|·,θ) (cid:55)→ (cid:37), where (cid:37) ∈ P and P is the space of functionals of the form θ (cid:55)→ p(·|·,θ) (cid:55)→ (cid:37). Thus, the prior is a function of the form ξ(θ,(cid:37)). But since the functional (cid:37) is a composite function with θ (cid:55)→ p(·|·,θ) (cid:55)→ (cid:37), ξ(θ,(cid:37)) is essentiallyafunctionofθ alone. Thus,weonlyuseξ(θ,(cid:37))notationwhenattentiontothesubsidiary computation p(·|·,θ) (cid:55)→ (cid:37) is required. Let q denote the proposal density. For a given θ, q(θ,θ∗) defines a distribution of potential new values θ∗. We use a move-one-at-a-time, random-walk, proposal density that puts its mass on discrete, separated points, proportional to a normal density. Two aspects of the proposal scheme are worth noting. The first is that the wider the separation between the points in the support of q the less accurately g(θ) needs to be computed for α at step 5 of the algorithm below to be correct. A practical constraint is that the separation cannot be much more than a standard deviation of the proposal density or the chain will eventually stick at some value of θ. Our separations are typically 1/2 of a standard deviation of the proposal density. In turn, the standard deviations of theproposaldensityaretypicallynomorethanthestandarddeviationsofthepriordistributionsof structural parameters shown in Tables 3 to 7 and no less than one order of magnitude smaller. The second aspect worth noting is that the prior is putting mass on these discrete points in proportion to ξ(θ). Because one does not have to normalize either the likelihood or the prior in an MCMC chain, normalization of densities does not matter for the computation of the chain and similarly for the joint distribution f(y|z,g(θ))ξ(θ) considered as a function of θ. However, f(y|z,ω) must be (cid:82) normalizedsuchthat f(y|x,ω)dy = 1toensurethattheimpliedmapexpressedin(7)iscomputed correctly. Thealgorithmfortheθ-chainisasfollows. Givenacurrentθo andthecorrespondingωo = g(θo), 21

we obtain the next pair (θ(cid:48),ω(cid:48)) as follows: 1. Draw θ∗ according to q(θo,θ∗). 2. Draw {yˆ,zˆ }N according to p(y |z ,θ∗). t t−1 t=1 t t−1 3. Compute ζ∗ = g(θ∗) and the functional (cid:37)∗ from the simulation {yˆ,zˆ }N . t t−1 t=1 (cid:16) (cid:17) L(θ∗)ξ(θ∗,(cid:37)∗)q(θ∗,θo) 4. Compute α = min 1, . L(θo)ξ(θo,(cid:37)o)q(θo,θ∗) 5. With probability α, set (θ(cid:48),ω(cid:48)) = (θ∗,ω∗), otherwise set (θ(cid:48),ω(cid:48)) = (θo,ωo). It is at step 3 that AG made an important modification to the algorithm proposed by GM. At that point one has putative pairs (θ∗,ω∗) and (θo,ωo) and corresponding simulations {yˆ∗,zˆ∗ }N and t t−1 t=1 {yˆo,zˆo }N . AG use ω∗ as a start and recompute ωo using the BFGS algorithm, obtaining ωˆo. If t t−1 t=1 N N (cid:88) (cid:88) logf(yˆo|zˆo ,ωˆo) > logf(yˆo|zˆo ,ωo), t t−1 t t−1 t=1 t=1 then ωˆo replaces ωo. In the same fashion, ω∗ is recomputed using ωo as a start. Once computed, a (θ,ω) pair is never discarded. Neither are the corresponding L(θ) and ξ(θ,(cid:37)). Because the support of the proposal density is discrete, points in the θ-chain will often recur, in which case g(θ), L(θ), and ξ(θ,(cid:37)) are retrieved from storage rather than computed afresh. If the modification just described results in an improved (θo,ωo), that pair and corresponding L(θo) and ξ(θo,(cid:37)o) replace the values in storage; similarly for (θ∗,ω∗). The upshot is that the values for g(θ) used at step 4 will be optima computed from many different random starts after the chain has run awhile. 3.3 GSM: Estimation of the auxiliary model The observed data are y for t = 1,...,n, where y is a vector of dimension M. The vector t t of observable variables used in estimation has four components: real equity returns, real interest rates, realpercapitaconsumptiongrowthrates, andrealdividendgrowthrates. ThesymbolsP,Q, V, etc. that appear in this section are general vectors (matrices) of statistical parameters and are not instances of the model parameters or functionals in Section 2. The data are modeled as y = µ +U ε t zt−1 zt−1 t 22

where µ = b +Bz , (8) zt−1 0 t−1 which is the location function of a k-lag vector auto-regressive (VAR(k)) specification, obtained by letting columns of B past the first kM be zero. In this formulation, U is the Cholesky factor of zt−1 Σ = U U(cid:48) (9) zt−1 0 0 +QΣ Q(cid:48) (10) zt−2 +P(y −µ )(y −µ )(cid:48)P(cid:48) (11) t−1 zt−2 t−1 zt−2 +max[0,V˜(y −µ )]max[0,V˜(y −µ )](cid:48), (12) t−1 zt−2 t−1 zt−2 where, as with B, the lag length is determined by letting the trailing columns of P and V˜ be zeros. In this application, the auxiliary model is not Markovian due to the recursion in expression (10).10 As in Gallant and Tauchen (2014), the last term in the model above captures the leverage effect. In computations, max(0,x) in expression (12), which is applied element-wise, is replaced by a twice differentiable cubic spline approximation that plots slightly above max(0,x) over (0.00,0.10) and coincides elsewhere. The density h(ε) of the i.i.d. ε is the square of a Hermite polynomial times a normal density, t the idea being that the class of such h is dense in Hellenger norm and can therefore approximate a density to within arbitrary accuracy in Kullback-Leibler distance, see Gallant and Nychka (1987). Such approximations are often called sieves; Gallant and Nychka term this particular sieve seminonparametric maximum likelihood estimator, or SNP.11 The density h(ε) is the normal when the degree of the Hermite polynomial is zero. In addition, the constant term of the Hermite polynomial can be a linear function of z . This has the effect of adding a nonlinear term to the location t−1 function (8) and the variance function (9). It also causes the higher moments of h(ε) to depend on z as well. The SNP auxiliary model is determined statistically by adding terms as indicated by t−1 the BIC protocol for selecting the terms that comprise a sieve, see Schwarz (1978). In our specification, U is an upper triangular matrix, P and V˜ are diagonal matrices, and Q 0 10See Gallant and Long (1997) for the properties of estimators of the form used in this section when the model is not Markovian. 11See Gallant and Tauchen (2014) for an introduction and implementation of the SNP estimation. 23

is scalar. The degree of the SNP h(ε) density is four. We specify that the constant term of the SNP density does not depend on the past. The auxiliary model chosen for our analysis, based on the BIC, has 1 lag in the conditional mean component, 1 lag in each of ARCH and GARCH terms. Although the univariate analysis of stock price dynamics generally incorporates a leverage term, we find in our SNP estimation with four variables that this term is not necessary according to the BIC. The auxiliary model in the SNP estimation has 51 parameters of which 50 are estimated and one determined by a normalization rule. The error distributions implied by the auxiliary model differ significantly from the distributions of innovation shocks assumed in those structural models in Section 2. We numerically solve the structural models assuming normally distributed innovation shocks to consumption and dividend growth rates. The error distributions of simulations from these models are markedly non-Gaussian. For example, in addition to GARCH effects, the fourdimensional error distribution implied by the AAMS model is skewed in all four components and has fat-tails for consumption growth, dividend growth and stock returns and thin tails for bond returns. 3.4 Relative model comparison Relative model comparison is standard Bayesian inference. The posterior probabilities of the five structuralmodelsmaybecomputedusingtheNewtonandRaftery(1994)pˆ4 methodforcomputing themarginallikelihoodfromanMCMCchainwhenassigningequalpriorprobabilitytoeachmodel. An alternative is method f of Gamerman and Lopes (2006), Section 7.2.1. The advantage of these 5 methods is that knowledge of the normalizing constants of the likelihood L(θ) and the prior ξ(θ) are not required. We do not know these normalizing constants due to the imposition of support conditions. Itisimportant,however,thattheauxiliarymodelbethesameforallmodels. Otherwise the normalizing constant of L(θ) would be required. One divides the marginal density for each model by the sum for all models to get the posterior probabilities for relative model assessment. Unfortunately, these and similar methods require that the range of the likelihoods that occur in the MCMC be within the float limits of the computing equipment employed. This can be remedied by left truncating the MCMC draws, which can be interpreted as a modification to the prior. 24

However, not only is it hard to interpret a truncation prior of this sort, but also we found that the implied ordering of the models is sensitive to the truncation for both the pˆ4 and f methods. 5 Therefore, in the results reported below we used the BIC for model selection. 3.5 The prior and its support Allstructuralmodelsconsideredinthispaperarerichlyparameterized. Werepresenttheparameter vector by θ. Table 2 summarizes structural parameters of all asset pricing models in Section 2. The prior of any structural parameter vector is the combination of the product of independent normal density functions and support conditions. The product of independent normal density functions is given by n˜ ξ(θ) = (cid:89) N (cid:2) θ | (cid:0) θ∗,σ2(cid:1)(cid:3) i i θ i=1 where n˜ denotes the number of parameters. The complete set of location and scale parameters for independentnormalpriorsaswellassupportconditionsareavailableintheInternetAppendix. We set the location parameter values such that the asset pricing models generate mean risk-free rate that is not too high and mean equity premium that is not close to zero. For all models’ parameters, we set the scale parameter values to be sufficiently large and use wide support intervals. This allows a wide range of parameter values of any model to be explored in the estimation, which in turn, provides ample room for asset pricing models to contribute to the identification of estimated parameters. Due to the support conditions, the effective prior is not an independence prior. For some values of θ∗ proposed in Step 1 of the θ-chain described in Section 3, a model solution at Step 2 may not exist. In such cases, α at Step 5 is set to zero. The prior support of the subjective discount factor (β), the coefficient of risk aversion (γ), and the EIS (ψ) parameter are set to 0.9 < β < 0.995, 0.1 < γ < 100, and 0.1 < ψ < 10, respectively. The subjective discount factor must be high enough to imply a reasonably low risk-free rate. The range 0.9 < β < 0.995 is wide compared to the prior on this parameter in Schorfheide et al. (2017). The support interval for γ that we use is much wider than the reasonable range 1 < γ < 10 suggested by Mehra and Prescott (1985). Different from calibration studies on long-run risks, we do not impose ψ > 1 but allow for possibilities of ψ < 1 and a preference for late resolution of uncertainty. For the ambiguity aversion parameter η, the support interval is γ < η < 200. Again, 25

this interval is wide given calibrated studies such as Ju and Miao (2012), Jahan-Parvar and Liu (2014) and Altug et al. (2017). Because the agent is ambiguity averse when η > γ, we impose this condition in estimating models with smooth ambiguity utility. The location parameters for β,γ,ψ and η in the prior are set at values consistent with the extant calibration studies. The scale parameters for these preference parameters are set to large values to deliver loose priors. For models EZMS, AAMS and AAMSSV, we use the parameter estimates and the associated standard errors reported in Cecchetti, Lam, and Mark (2000) to determine the location and scale parameter values for parameters µ ,µ ,σ ,p and p in the Markov-switching model of consumph l c hh ll tiongrowth. IntheAAMSSVmodelwithtime-varyingvolatility,ourparameterchoicesforlocation and scale of pσ , pσ,σ and σ rely on estimates of Lettau et al. (2008) and Boguth and Kuehn hh ll h l (2013). The location values of the dividend volatility parameter σ and the leverage parameter d λ are determined by the calibration of Ju and Miao (2012). Following Abel (1999), we impose λ ≥ 1 in the estimation. Estimation results of Bansal et al. (2007), Aldrich and Gallant (2011), and Schorfheide et al. (2017) lead to values of λ in the [1.5,4.5] range. We choose 1 ≤ λ ≤ 6 as the support interval. For models AALRRSV and EZLRRSV, we use the calibrated parameter values in Bansal et al. (2012) and priors postulated in Schorfheide et al. (2017) to choose the location and scale parameter values, and support intervals as well. For example, the location of the unconditional mean of consumption growth, µ , is set at 0.02 with a small scale parameter value. The location of the c persistenceparameterofthelong-runriskcomponent, ρ , issetat0.95withalargescaleparameter x value of 0.2. The support interval for ρ is −0.99 < ρ < 0.99. Similarly, other model parameters x x also have loose priors and wide support intervals as in Schorfheide et al. (2017). 3.6 Estimation results We summarize estimation results using the GSM method in Tables 3 to 7.12 We plot the prior and posterior densities of the estimated structural parameters in Figures 3 to 7. These plots show considerable shifts in both location and scale between priors and posteriors, suggesting that the estimation procedure and data have a significant impact on the estimation results. The impact of 12Foreachassetpricingmodel,werunthestandardMCMCchainwiththelikelihoodputto1ateverydrawtoobtain the prior distribution of model parameters presented in Tables 3 to 7 and Figures 3 to 7. 26

priors and support conditions is notable, but of second order of importance. Estimation results show that the posterior estimates of β are tightly bounded in all models and generallyimplylowrisk-freerates. ThereisanongoingdebateaboutthevalueoftheEISparameter (ψ) in the macro-finance literature. This parameter is crucial for equilibrium asset pricing models to match macroeconomic and financial moments in the data, see Bansal and Yaron (2004), Croce (2014)andLiuandMiao(2015)amongothers. Somestudies(e.g., Hall,1988andLudvigson,1999) find that the EIS estimate is less than 1, based on aggregate consumption and asset returns data. Other studies find higher values using cohort- or household-level data (e.g., Attanasio and Weber, 1993 and Vissing-Jorgensen, 2002). Our estimation strongly suggests an EIS greater than 1 and thus a preference for early resolution of uncertainty. As shown in Tables 3 to 7, the posterior mean, median, 5and95percentilesofψ estimatesareallabove1inallmodelspresentedinSection2. The plots of the posterior density for ψ in Figures 3 to 7 also reveal that the posterior dispersion of this parameter over the MCMC chain is small. Jeong et al. (2015) estimate the recursive multiple prior utility model using asset prices data and obtain estimates of ψ greater than 10. High estimates of ψ generated from our estimation imply low and stable risk-free rates (see Section 4). In a DSGE analysis with broader scope, Bianchi et al. (2016) rely on the mechanism of time-varying ambiguity on operating costs to ease the tension between excess equity volatility and smooth risk-free rates. The posterior estimates of ψ for models AAMS and EZMS are high and comparable to the estimates in the long-run risk literature. The posterior mean, median and 5 and 95 percentiles of ψ estimates are moderately higher in the EZMS model than in the AAMS model, with the posterior mean and median being above 2. The ψ estimates in the EZMS model are close to results obtained by Schorfheide et al. (2017) and Bansal, Kiku, and Yaron (2016). Our estimation results suggest that incorporating ambiguity in the model leads to lower estimates of ψ. This is also evident from a comparison of estimates in the EZLRRSV model and in the AALRRSV model. The posterior estimates of ψ are significantly lower in a long-run risk model with ambiguity than in a pure longrun risk model. Nevertheless, our estimates of ψ in the long-run risk model are still lower than thosereportedbySchorfheideetal.(2017). Thediscrepancyarisesbecause1)weusetheprojection method rather than log-linear approximation to solve models, 2) we use the GSM Bayesian method for model estimation, and 3) we use a different sample of data. 27

Our estimation results strongly support asset pricing models with smooth ambiguity. The posterior estimates of the ambiguity aversion parameter η are significantly large in models AAMS, AAMSSV and AALRRSV. Not surprisingly, the estimates obtained for the AAMS model are close to the calibrated value in Ju and Miao (2012) (η = 8.864). In addition, the estimates of η are modestly higher when regime-switching volatility in consumption growth is incorporated in the estimation. We observe that the posterior mean and median of η are about 10 in the AAMSSV modelwhileabout7intheAAMSmodel. Inthelong-runrisksetting,theGSMBayesianestimation generates high posterior estimates of η with mean and median of about 23. These results suggest that empirical support for models with smooth ambiguity is robust to different specifications of consumptiondynamicsandthattheextentofambiguityaversionlargelydependonotherpreference parameters and primitive parameters in the consumption and dividend growth processes. While the estimated degree of ambiguity aversion varies across several models, these estimates are all reasonable from the perspective of decision-making. One could conduct thought experiments as in Halevy (2007) and Ju and Miao (2012) to gauge reasonable values of the ambiguity aversion parameter. Estimates of the coefficients of risk aversion γ importantly hinge on the presence of ambiguity aversion. Estimation results of models EZMS and EZLRRSV show that the posterior mean and medianofγ arehighandthe5and95percentilesimplytightboundsfortheestimate. Inparticular, the posterior estimates of γ in the estimated long-run risk model EZLRRSV are close to the results reported by Schorfheide et al. (2017) and Bansal et al. (2016). The posterior mean of γ is 8.4, and the associated 95 percentile value is 10.4. These values are also close the the calibrated values in Bansal and Yaron (2004) and Bansal et al. (2012). On the other hand, the γ estimate is more dispersed in models with smooth ambiguity, i.e., models AAMS, AAMSSV and AALRRSV, as is evident from wide (5%, 95%) intervals. In a related work, Chen, Favilukis, and Ludvigson (2013) estimate preference parameters of recursive utility using a semiparametric technique. Their estimated relative risk aversion parameter ranges from 17 to 60. IntheGSMBayesianestimation,primitiveparametersintheconsumptionanddividendgrowth processes are jointly estimated with preference parameters. Models AAMS, AAMSSV and EZMS have Markov-switching consumption growth while models AALRRSV and EZLRRSV feature long- 28

run risks. In the Markov-switching environment, our estimation method identifies a normal regime and a contraction regime for mean consumption growth. The posterior estimates of µ are largely h in line with the historical average annual consumption growth. For instance, the posterior mean and median of µ in the AAMS model are about 2%. In addition, the posterior estimates of the h transition probabilityp (pµ in model AAMSSV) are close to 1 and thus indicate thatthis regime hh hh is very persistent. Furthermore, the estimates of low mean growth regime for these models indicate a relatively transitory contraction regime with lower estimates of the transition probability p (pµ ll ll in model AAMSSV). Note that we obtain these estimates from structural estimation of asset pricing models using data on both fundamentals and asset returns. The GSM Bayesian estimation takes into account equilibriumassetpricesandyieldsestimatedconsumptiondynamicsthatcorrespondstotheagent’s subjective belief. Compared with estimates of the parameters of the Markov-switching model reportedbycalibrationstudies(e.g., Cecchettietal.(2000)andJuandMiao(2012)), ourestimates imply a “peso” version of the model. That is, the severe contraction state rarely occurs in the observed data or simulations due to its low likelihood (1 − p ). However, because an agent hh cannot observe the mean growth state and is also aware of severity (µ ) and persistence (p ) of the l ll contractionregime, theagentisalwaysconcernedaboutstateuncertaintyandmoreover, ambiguity aversion magnifies the impact of this concern. In addition, the posterior estimates of the low mean regimeµ seemtoolowgiventhepost-warexperienceoftheeconomy,andtheestimatedpersistence l of this regime varies significantly across different models. These results suggest that apart from ambiguity on the mean growth state, extra sources of ambiguity about parameters of the Markovswitching model may co-exist. In estimating the AAMSSV model, we find two distinct volatility regimes, both of which are persistent. This result is consistent with the findings of Lettau et al. (2008) and Boguth and Kuehn (2013). However, the posterior estimates of the high volatility regime σ are too high to h be reconciled with the post-war consumption data. The estimates of µ are even more negative l than the estimates for the AAMS model. Nevertheless, these estimates are more consistent with the long sample of Shiller’s data.13 Again, extra sources of ambiguity may arise due to learning 13WethankRobertShillerformakingthedataavailableathttp://www.econ.yale.edu/∼shiller/data/chapt26.xlsx. 29

from past experiences or parameter uncertainty.14 For models AAMS and AAMSSV, the leverage parameter λ and the dividend growth volatility σ estimates are reasonably close to the calibrated d values considered by Abel (1999), Bansal and Yaron (2004) and Ju and Miao (2012). The posterior estimates of λ are roughly between 2 and 4 with a posterior mean of about 3 for both models. The estimates of λ and σ for the EZMS model are significantly higher than those for models AAMS d and AAMSSV. Turning to estimation results of models featuring long-run risks, we find that the estimated models AALRRSV and EZLRRSV both provide support to the presence of a persistent component in the consumption growth process. This empirical support is evident even when ambiguity about conditional mean growth is incorporated in the model. The posterior estimates of the persistence parameterρ arecloseto1withnarrow(5%,95%)intervals. Convertedintoestimatesatamonthly x frequency, our results are similar to those reported by Schorfheide et al. (2017). In addition, the stochastic volatility component is also persistent in our estimation, a result consistent with Schorfheide et al. (2017).15 Other parameter estimates including µ , µ , σ λ, φ and φ are c s w d c similar to the estimates reported by the studies on long-run risks such as Bansal et al. (2012), Bansal et al. (2016) and Schorfheide et al. (2017). We present results of relative model comparison in Tables 3 to 7, based on the maximum of the log likelihood and the BIC for all estimated models. We use the auxiliary model presented in Section 3.3 and the MCMC chain of structural parameters of each asset pricing model to compute themaximumoftheloglikelihoodandtheBICofthemodel. Accordingtothesetwocriteria,among all five estimated models the AAMSSV best characterizes the data in that the model provides the best fit of the SNP density given the observed data. The log likelihood computation leads to the model ranking AAMSSV(cid:31)AALRRSV(cid:31)EZLRRSV(cid:31)EZMS(cid:31)AAMS. The BIC gives us the same ranking except that EZMS(cid:31)EZLRRSV because the number of model parameters is also taken into account. Based on the BIC ranking the AALRRSV model is “close” to AAMSSV, but the remainder are more than 40 orders of magnitude distant. These findings suggest that (1) time- 14A full-fledged analysis of modeling multiple sources of ambiguity requires development of new models that have parameter uncertainty, state uncertainty and learning. Estimating this class of models is beyond the scope of our current study. 15ApplyingtheGSMBayesianestimation,wefindthattheparametervalueofρ intheMCMCchainremainsstagnant s at a high level (ρ =0.95). s 30

varying volatility in consumption is important for asset pricing models to deliver the SNP densities that fit the data well, because according to the log likelihood criterion priority is given to models AAMSSV, AALRRSV and EZLRRSV, all of which feature time-varying volatility, and (2) asset pricing models (AAMSSV and AALRRSV) with ambiguity, learning and time-varying volatility are preferred to the long-run risk model EZLRRSV in the statistical model comparison. Although the model of Ju and Miao (2012), AAMS, receives less statistical support than other models do, it can match key financial moments well, as shown in the next section. 4 Asset Pricing Implications 4.1 Variance risk premium The moments of equity returns are naturally defined under the physical measure implied by fundamentals and the state variables in any asset pricing model. Furthermore, we can study the dynamics of the risk-neutral variance and variance risk premium (henceforth, VRP) generated from models considered above. As noted in Bollerslev, Tauchen, and Zhou (2009), the market variance risk premium is defined as the difference between the expected equity return variances under the risk-neutral and physical measures, and it measures the risk premium compensation for investors bearing the variance risk. Several studies show that the mean and volatility of the market variance risk premium are high, which poses a serious challenge to many existing asset pricing models, for example, see the discussion in Drechsler (2013). In a calibration study, Miao, Wei, and Zhou (2012) find that the AAMS model can roughly match the mean and volatility of the VRP in the data. Here, we take a different stance in that we do not calibrate any model to target moments of the VRP. Instead, we examine whether our estimated models produce empirically reasonable dynamics of the VRP. In the literature, a commonly used empirical proxy for the risk-neutral volatility is the Chicago Board Options Exchange (CBOE)’s volatility index (VIX). In the empirical analysis, we measure the market variance risk premium as the difference between the model-free implied variance and the conditional projection of realized variance. Our empirical estimation of the VRP closely follows the study of Liu and Zhang (2015), which applies the CBOE’s methodology of constructing the 31

VIX to index options with 90 days maturity. To estimate the variance of equity returns under the physical measure, we first compute realized returns and then take a linear projection to obtain the conditional variance, which denoted by VOL2. The variance risk premium is defined as t VRP = VIX2−VOL2. t t t In the model, the risk-neutral variance VIX2 takes the form t E (cid:2) M σ2 (cid:3) VIX2 = EQ(cid:2) σ2 (cid:3) = t t,t+1 r,t+1 t t r,t+1 E [M ] t t,t+1 where Q denotes the risk-neutral measure, and the expected variance under the physical measure is given by VOL2 = E (cid:2) σ2 (cid:3) t t r,t+1 where σ2 = E (cid:2) r2 (cid:3) −(E [r ])2. r,t t t+1 t t+1 4.2 Impulse responses We perform impulse responses analyses for the estimated asset pricing models by investigating key financial variables including the SDF, price-dividend ratio, conditional equity premium, equity volatility and variance risk premium. We use mean estimates reported in Tables 3 to 7 to parameterize models and compute impulse responses functions. Results for models AAMS, AAMSSV, AALRRSV and EZLRRSV are plotted in Figures 8 and 9. We assume that the shock to mean growth rate of consumption occurs in the third period and lasts only one period. Figure8showsthatwhenthemeanconsumptiongrowthregimeshiftsfrom“high”(µ )to“low” h (µ ), Bayesian updating leads to a lower level of belief π . Veronesi (1999) has shown that with l t CRRA utility, the impact will be an increase in conditional equity volatility and equity premium. This effect is amplified under ambiguity aversion. The plotted ambiguity-distorted belief manifests endogenouspessimismthatimpliesasharpincreaseintheSDFandadecreaseintheprice-dividend ratio. As a result, the conditional equity volatility and equity premium rise significantly. Since conditional volatility rises in states where the SDF is high, the risk-neutral variance increases 32

more than the physical return variance does, leading to an increase in the VRP. Figure 8 displays qualitatively similar impulse responses of beliefs and financial variables for the AAMSSV model where the consumption volatility state is assumed to be σ throughout the response periods.16 The h notable discrepancies in the magnitude of responses between the AAMS model and the AAMSSV modelarelargelyduetotheinclusionoftime-varyingvolatilityintheAAMSSVmodelanddifferent parameter estimates as discussed in Section 3.6. Figure 9 displays the responses of key variables in models AALRRSV and EZLRRSV when a negative shock of size −4ϕ µ hits the long-run risk component x , which is assumed to be zero x s t initially. Different from the AAMS model with Markov-switching growth rates, in the AALRRSV model Bayesian filtering of x implies persistent movements in financial variables because of its t long-run risk feature. Again, the plotted ambiguity-distorted belief reflects the agent’s pessimistic view about the conditional mean growth rate of consumption. In line with the long-run risk model, learning about x produces a SDF and a price-dividend ratio that move in the opposite directions t upon the impact of the shock. Thus, in the AALRRSV model the long-run risk component carries a positive risk premium. Because the conditional volatility of consumption growth is assumed to be constant in this analysis, the conditional equity volatility decreases on impact and rises slowly afterwards. Theconditionalequitypremiumexhibitsasimilarresponseasaconsequence. TheVRP falls at first and rises afterwards, due to the response of the conditional equity volatility. Figure 9 shows similar impulse responses for the EZLRRSV model in which the long-run risk component is fully observable. In both models, the response of the VRP is negligible compared to the results for models AAMS and AAMSSV. 4.3 Financial moments We investigate the ability of all estimated models in replicating unconditional moments of key macroeconomic and financial variables. Unlike calibration studies, our aim is not to match unconditional moments of asset returns in the data as closely as possible. Instead, we assess the impact of ambiguity aversion on financial variables based on estimated parameter values. In addition, we examinehowwellourestimatedmodelscanmatchmomentsofassetreturns, giventhatourestima- 16The impulse responses plot for the EZMS model is similar and thus omitted here for the sake of brevity. 33

tion strategy is designed not to match moments but to fit the SNP densities of asset pricing models given the observed data. If any estimated model is reasonably successful in reproducing unconditional moments of consumption growth and asset returns, we view this outcome as confirmation that the model characterizes the underlying data generating process well. This analysis makes our structural estimation more relevant from an alternative empirical perspective. By examining asset pricing implications of estimated models, our analysis supersedes previous studies on structural estimations such as Bansal et al. (2007), Aldrich and Gallant (2011) and Jeong et al. (2015). Table8presentsunconditionalmomentsofassetreturnssimulatedfromallassetpricingmodels considered in this paper. For each model, we compute these moments on a MCMC chain of 12,000 estimates and report mean, median, standard deviation, 5th and 95th percentiles of simulation results. To facilitate comparison, we present moments computed from the historical U.S. data. Due to the high EIS estimates in all models and resulting intertemporal substitution effect, the meanandvolatilityoftherisk-freeratearelowacrossthesemodels. Allmodelsproducesimulations on their chains of estimates that contain the historical equity premium and return volatility in the (5%,95%) intervals. Table 8 shows that among all models, the AAMS model can best match moments of returns. The estimated AAMS model delivers mean and volatility of the risk-free rate, equity premium and return volatility, and mean and volatility of the VRP close to the moments computed from the data. In addition, the 5th and 95th percentiles of simulated moments are sufficiently tight to include the data moments except for the volatilities of the risk-free rate and VRP. The intuition of the impact of ambiguity on asset returns has been illustrated in previous studies, for example see Ju and Miao (2012) and Collard et al. (2017). That is, the precautionary savings motive driven by ambiguity aversion reduces the risk-free rate, and in addition to the standard risk premium the agent also demands an uncertainty premium for being ambiguous about the data-generating process. The latter mechanism is evident from inspecting the market price of risk, which is defined as σ(M )/E(M ). According to the conditional version of the Euler equation: t,t+1 t,t+1 σ (M ) E (R )−R = − t t,t+1 σ (R )ρ (M ,R ), t t+1 f,t E (M ) t t+1 t t,t+1 t+1 t t,t+1 34

the high market price of risk implied by the AAMS model leads to a high equity premium. Since the estimated model also produces volatility of dividend growth close to the data and the leverage parameterconsistentwithpreviouscalibrationstudies,themodelcannaturallymatchthevolatility of equity returns in the data. The AAMS model also generates a high VRP close to the data. This is a remarkable result, since we do not use the risk-neutral variance data to aid estimation. The implied high VRP is a consequence of strong co-movement of the SDF and the return volatility when the economy shifts to a bad state. The co-movement therefore leads to a substantial wedge between the risk-neutral variance and the objective variance. On the other hand, the estimated EZMS model (ambiguity neutral case) shows poor performance in matching the moments. The mean of simulated equity premium in this model is only half of the historical equity premium whereas the moments of the VRP are much higher than the data. It is evident in Table 8 that incorporating time-varying consumption volatility in the Markovswitching model does not yield significantly better asset pricing results, though the GSM Bayesian estimation provides statistical support to this model relative to the more parsimonious model AAMS. The model predicts mean values of equity premium and VRP moderately higher than the data. The range of the 5th−95th percentile is wider than that in the AAMS model both for the simulated equity premium and the VRP. The mean of the market price of risk increases greatly with the addition of regime-switching conditional volatility. In the long-run risk setting, the equity premium and market price of risk implied by the AAL- RRSVmodelishigherthanthoseintheEZLRRSVmodelduetothesignificantimpactofambiguity. However, neither model is able to match moments of the VRP in the data. Both models generate mean and volatility of the VRP close to zero. This is in contrast to models AAMS and AAMSSV that can match both equity premium and mean VRP well. In fact, one must introduce jumps in state processes to generate a high and volatile VRP in the long-run risk setting, for example see Drechsler (2013). We leave structural estimation of this class of models for future research. As the AAMS model can best match unconditional moments of financial variables, we next study conditional financial moments generated by this model. Because the AAMSSV model is an extension of AAMS and a statistically preferred model as suggested by the model comparison, we 35

also investigate conditional financial moments in AAMSSV. Figure 10 shows simulated conditional equity premium, return volatility, market price of risk and the VRP plotted against the state variable π in model AAMS. The conditional moments are drawn from the 5th to 95th percentile of the t simulations implied by 12,000 MCMC estimates of structural parameters of the model. We also showconditionalmomentsgeneratedfromJuandMiao(2012)’scalibrationforcomparison. Weobservethatthesimulated90%regionofconditionalmomentsdoesnotincludethecalibrationresults of Ju and Miao (2012). This is because Ju and Miao (2012) use a long sample for their calibration. Figure 11 plots simulated conditional moments for model AAMSSV, where in each simulation the expectation with respect to volatility states is computed using stationary probabilities of the two volatility regimes. For both models, we observe that the key conditional financial moments exhibit a hump-shape when plotted against π , and that conditional equity premium, market price of risk t and VRP peak when π values are high. Our estimation implies a very persistent normal regime t for consumption growth with p close to 1, which leads to thisresult. Suppose that the economy hh initially stays in the normal regime. A negative shock to consumption prompts the agent to update his belief π downward, leading to enhanced state uncertainty. Ambiguity aversion further exact erbates the scenario by inducing endogenous pessimism and thus implies a significant increase in conditional equity premium, market price of risk and VRP. 5 Conclusion We have estimated a series of consumption-based asset pricing models with and without smooth ambiguity preferences. We use the GSM Bayesian estimation method developed by Gallant and McCulloch (2009) and an encompassing and flexible auxiliary model to jointly estimate preference parameters and dynamic models of consumption and dividend growth postulated in asset pricing models. We employ the semi-nonparametric method to estimate the auxiliary model and the GSM Bayesian method to obtain posterior estimates of structural parameters of asset pricing models. Our structural estimation with macro-finance data provides statistical support to asset pricing models with smooth ambiguity. Based on our estimation results, the quantitative effects of smooth ambiguity on asset returns are significant, both in the Markov-switching and long-run risk environments. 36

Our main findings are: (1) the distinction between risk aversion and ambiguity aversion is robust to the intertemporal substitution effect, i.e., our estimation provides statistical support to an ambiguity-averse representative agent who also prefers early resolution of uncertainty (or has a high EIS), (2) the statistical support for smooth ambiguity is robust to specifications of consumption and dividend processes, (3) a model comparison shows that models with ambiguity, learningandtime-varyingvolatilityarepreferredtothelong-runriskmodel,and(4)intheMarkovswitching environment our estimation identifies a normal regime and a contraction regime for the mean growth rate of consumption as well as two distinct volatility regimes; in the long-run risk environment our estimation identifies the long-run risk component. 37

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Table 1: Summary Statistics of the Data re rf re−rf ∆c ∆d t t t t t t Mean 5.98 0.96 5.03 1.83 1.56 St. Dev. 19.70 2.47 19.96 2.14 14.08 Skewness -0.8193 -1.4763 -0.6988 0.1079 -0.8716 Kurtosis 0.5926 5.0291 0.4457 0.0360 2.8810 J-B Test 0.0135 0.0010 0.0263 0.5000 0.0010 This table reports summary statistics for annual U.S. data (1941–2015). Mean and standard deviations of aggregate equity returns (r ), one-year Treasury Bill rate (rf), excess returns (r −rf), real per capita log t t t t consumption growth (∆c ), and real log dividend growth (∆d ) are expressed in percentages. “J-B test” t t reports the p-values of Jarque and Bera (1980) test of normality, where the null hypothesis is that the time series is normally distributed. Table 2: Model Summary Model State variables Parameters AAMS π {β,γ,ψ,η,µ ,µ ,p ,p ,σ,λ,σ } t h l hh ll d AAMSSV (π , sσ) {β,γ,ψ,η,µ ,µ ,pµ ,pµ,σ ,σ ,pσ ,pσ,λ,σ } t t h l hh ll h l hh ll d AALRRSV (xˆ , ν , σ ) {β,γ,ψ,η,µ ,ρ ,ϕ ,λ,ϕ ,µ ,ρ ,σ } t t t c x x d σ σ w EZMS π {β,γ,ψ,µ ,µ ,p ,p ,σ,λ,σ } t h l hh ll d EZLRRSV (x ,σ2) {β,γ,ψ,µ ,ρ ,ϕ ,λ,ϕ ,ϕ ,µ ,ρ ,σ } t t c x x d c σ σ w This table summarizes relevant state variables and structural parameters for each asset pricing model described in Section 2. 42

Table 3: GSM Estimation Results: the AAMS Model Prior Posterior Parameter Mean Median 5% 95% Mean Median 5% 95% β 0.985 0.985 0.978 0.993 0.975 0.974 0.969 0.985 γ 4.908 4.750 3.250 6.750 2.841 3.063 0.563 4.563 ψ 1.512 1.563 1.188 1.813 2.040 2.031 1.781 2.406 η 9.109 9.500 6.500 12.500 6.959 6.938 5.063 8.938 p 0.543 0.531 0.344 0.781 0.835 0.839 0.786 0.888 ll p 0.783 0.813 0.563 0.938 0.996 0.997 0.994 0.997 hh µ -0.059 -0.059 -0.074 -0.035 -0.039 -0.039 -0.048 -0.031 l µ 0.022 0.021 0.014 0.033 0.022 0.022 0.016 0.029 h λ 2.598 2.750 1.250 3.750 3.420 3.422 2.703 4.203 σ 0.028 0.029 0.018 0.041 0.019 0.019 0.015 0.022 c σ 0.137 0.133 0.086 0.180 0.137 0.137 0.113 0.168 d BIC 832.14 Log likelihood -392.32 MCMC repetitions 10,000 12,000 Thistablepresentspriorandposteriormarginalmeans,medians,5and95percentilesofmodelparametersfor theAAMSmodel. “BIC”representstheBayesianinformationcriteria,seeSchwarz(1978). “Loglikelihood” representsthemaximumoftheloglikelihoodoftheencompassingmodelovertheMCMCchainofestimates. MCMCrepetitionsaftertransientshavedissipatedarereportedforboththepriorandposterior. Estimation results are for the U.S. annual data 1941–2015. 43

Table 4: GSM Estimation Results: the AAMSSV Model Prior Posterior Parameter Mean Median 5% 95% Mean Median 5% 95% β 0.984 0.983 0.978 0.991 0.982 0.984 0.972 0.991 γ 4.723 4.750 3.250 6.250 1.167 0.875 0.125 4.125 ψ 1.483 1.438 1.188 1.813 1.357 1.348 1.090 1.668 η 9.235 9.500 6.500 12.500 10.252 10.125 6.875 13.625 pµ 0.508 0.531 0.281 0.719 0.668 0.686 0.504 0.746 ll pµ 0.806 0.813 0.563 0.938 0.996 0.998 0.984 0.999 hh µ -0.059 -0.059 -0.074 -0.027 -0.056 -0.057 -0.068 -0.042 l µ 0.022 0.021 0.014 0.029 0.023 0.023 0.014 0.033 h pσ 0.849 0.859 0.734 0.953 0.986 0.990 0.948 0.996 ll pσ 0.841 0.859 0.734 0.953 0.982 0.984 0.957 0.995 hh σ 0.015 0.015 0.009 0.021 0.013 0.012 0.004 0.022 l σ 0.030 0.029 0.018 0.041 0.038 0.038 0.029 0.050 h λ 2.881 2.750 1.750 3.750 2.739 2.641 1.953 4.016 σ 0.131 0.133 0.086 0.180 0.159 0.157 0.122 0.210 d BIC 746.31 Log likelihood -342.93 MCMC repetitions 10,000 12,000 This table presents prior and posterior marginal means, medians, 5 and 95 percentiles of model parameters for the AAMSSV model. “BIC” represents the Bayesian information criteria, see Schwarz (1978). “Log likelihood” represents the maximum of the log likelihood of the encompassing model over the MCMC chain ofestimates. MCMCrepetitionsaftertransientshavedissipatedarereportedforboththepriorandposterior. Estimation results are for the U.S. annual data 1941–2015. 44

ledoM VSRRLAA eht :stluseR noitamitsE MSG :5 elbaT roiretsoP roirP %59 %5 naideM naeM %59 %5 naideM naeM retemaraP 299.0 979.0 789.0 689.0 989.0 679.0 189.0 289.0 β 604.6 917.1 130.5 386.4 526.6 526.2 578.4 197.4 γ 587.1 210.1 311.1 522.1 187.1 651.1 964.1 284.1 ψ 005.53 005.01 005.32 173.32 000.33 000.71 000.52 629.42 η 020.0 710.0 910.0 910.0 120.0 710.0 910.0 910.0 µ c 759.0 629.0 149.0 149.0 969.0 135.0 187.0 277.0 ρ x 592.0 791.0 842.0 842.0 722.0 680.0 841.0 751.0 φ x 276.4 359.2 354.3 555.3 573.4 526.1 578.2 509.2 λ 448.5 448.3 609.4 778.4 521.4 573.1 526.2 457.2 φ d 120.0 910.0 020.0 020.0 820.0 110.0 120.0 020.0 µ s 059.0 059.0 059.0 059.0 969.0 969.0 969.0 969.0 ρ s 40-E97.2 40-E73.2 40-E45.2 40-E75.2 40-E15.3 40-E73.1 40-E92.2 40-E73.2 σ w 90.567 CIB 46.653doohilekil goL 000,21 000,01 snoititeper CMCM ”CIB“ .ledom VSRRLAA eht rof sretemarap ledom fo selitnecrep 59 dna 5 ,snaidem ,snaem lanigram roiretsop dna roirp stneserp elbat sihT gnissapmocneehtfodoohilekilgolehtfomumixamehtstneserper”doohilekilgoL“ .)8791(zrawhcSees,airetircnoitamrofninaiseyaBehtstneserper .roiretsop dna roirp eht htob rof detroper era detapissid evah stneisnart retfa snoititeper CMCM .setamitse fo niahc CMCM eht revo ledom .5102–1491 atad launna .S.U eht rof era stluser noitamitsE 45

Table 6: GSM Estimation Results: the EZMS Model Prior Posterior Parameter Mean Median 5% 95% Mean Median 5% 95% β 0.985 0.985 0.978 0.991 0.976 0.976 0.970 0.986 γ 4.771 4.750 3.250 6.250 2.909 2.906 2.344 3.484 ψ 1.488 1.438 1.188 1.813 2.400 2.281 1.844 3.656 p 0.530 0.531 0.281 0.781 0.972 0.972 0.943 0.989 ll p 0.774 0.813 0.563 0.938 0.993 0.993 0.987 0.999 hh µ -0.059 -0.059 -0.074 -0.035 -0.030 -0.029 -0.042 -0.017 l µ 0.022 0.021 0.014 0.029 0.030 0.030 0.020 0.041 h λ 2.647 2.750 1.750 3.750 4.974 5.109 3.391 5.859 σ 0.028 0.029 0.018 0.037 0.021 0.022 0.010 0.029 c σ 0.134 0.133 0.086 0.180 0.181 0.184 0.137 0.223 d BIC 812.98 Log likelihood -384.90 MCMC repetitions 10,000 12,000 Thistablepresentspriorandposteriormarginalmeans,medians,5and95percentilesofmodelparametersfor the EZMS model. “BIC” represents the Bayesian information criteria, see Schwarz (1978). “Log likelihood” representsthemaximumoftheloglikelihoodoftheencompassingmodelovertheMCMCchainofestimates. MCMCrepetitionsaftertransientshavedissipatedarereportedforboththepriorandposterior. Estimation results are for the U.S. annual data 1941–2015. 46

ledoM VSRRLZE eht :stluseR noitamitsE MSG :7 elbaT roiretsoP roirP %59 %5 naideM naeM %59 %5 naideM naeM retemaraP 989.0 779.0 289.0 289.0 199.0 879.0 589.0 489.0 β 834.01 912.6 135.8 134.8 057.6 052.3 057.4 879.4 γ 711.2 722.1 857.1 237.1 318.1 360.1 834.1 844.1 ψ 120.0 810.0 910.0 910.0 020.0 710.0 910.0 910.0 µ c 269.0 368.0 819.0 809.0 839.0 834.0 318.0 267.0 ρ x 542.0 541.0 481.0 981.0 722.0 680.0 841.0 151.0 φ x 229.3 745.2 141.3 761.3 052.4 057.1 052.3 779.2 λ 443.5 198.3 495.4 206.4 052.4 057.1 057.2 648.2 φ d 745.3 823.1 792.2 553.2 052.4 057.1 057.2 529.2 φ c 220.0 020.0 120.0 120.0 820.0 310.0 120.0 120.0 µ s 059.0 059.0 059.0 059.0 839.0 839.0 839.0 839.0 ρ s 40-E64.2 40-E31.2 40-E13.2 40-E82.2 40-E15.3 40-E86.1 40-E95.2 40-E55.2 σ w 20.518 CIB 16.183doohilekil goL 000,21 000,01 snoititeper CMCM ”CIB“ .ledom VSRRLZE eht rof sretemarap ledom fo selitnecrep 59 dna 5 ,snaidem ,snaem lanigram roiretsop dna roirp stneserp elbat sihT gnissapmocneehtfodoohilekilgolehtfomumixamehtstneserper”doohilekilgoL“ .)8791(zrawhcSees,airetircnoitamrofninaiseyaBehtstneserper .roiretsop dna roirp eht htob rof detroper era detapissid evah stneisnart retfa snoititeper CMCM .setamitse fo niahc CMCM eht revo ledom .5102–1491 atad launna .S.U eht rof era stluser noitamitsE 47

Table 8: Financial Moments E(rf) σ(rf) E(r −rf) σ(r −rf) E(VRP ) σ(VRP ) MPR t t t t t t t t Data 1.41 2.82 5.32 17.77 11.07 24.94 N.A. AAMS Mean 1.595 1.541 5.812 18.441 12.955 9.300 1.280 Median 1.399 1.598 6.349 18.266 12.644 8.833 1.308 Std 0.800 0.286 1.738 2.220 2.967 2.720 0.341 95% 2.941 1.951 8.033 22.729 18.860 14.495 1.819 5% 0.444 1.951 3.010 15.378 8.332 5.411 0.758 AAMSSV Mean 1.183 1.680 6.371 22.818 17.090 14.007 2.987 Median 1.267 1.673 5.910 22.579 14.678 10.430 2.790 Std 0.959 0.580 3.542 4.469 12.092 13.418 1.414 95% 2.465 2.685 14.446 30.242 35.407 39.337 5.970 5% -0.622 0.800 1.650 15.633 4.835 3.878 1.205 AALRRSV Mean 1.079 1.555 7.841 21.134 -0.744 1.752 1.312 Median 1.095 1.581 7.817 21.103 -0.288 1.086 1.142 Std 0.473 0.259 1.853 3.629 1.263 1.782 0.772 95% 1.818 1.946 10.838 28.345 0.406 6.490 2.976 5% 0.226 1.128 4.863 16.106 -3.218 0.438 0.489 EZMS Mean 1.282 2.100 2.919 40.489 58.704 67.854 0.698 Median 1.278 2.035 2.727 41.471 54.164 65.354 0.647 Std 0.468 0.394 1.634 8.211 26.877 35.106 0.208 95% 2.107 2.902 6.007 52.464 107.602 128.264 1.128 5% 0.511 1.579 0.706 28.070 21.067 21.228 0.454 EZLRRSV Mean 1.708 0.973 4.318 17.633 1.436 0.549 0.569 Median 1.686 0.856 4.458 17.560 1.398 0.524 0.573 Std 0.336 0.385 1.150 1.906 0.396 0.280 0.078 95% 2.289 1.589 5.999 20.903 2.137 1.130 0.692 5% 1.169 0.718 2.337 14.413 0.858 0.200 0.428 Thistablepresentsunconditionalfinancialmomentsgeneratedfromtheestimatedmodels. Thesequantities are computed from simulated variables paths based on 12,000 Bayesian MCMC estimates of the structural parameters. E(rf)andE(r −rf)aremeanrisk-freerateandmeanequitypremiumrespectively(inpercentt t t age). σ(rf) and σ(r −rf) are volatilities of risk-free rates and excess returns respectively (in percentage). t t t Moments of asset returns are computed based on annual data for the period 1941–2015. Variance risk premium (VRP) data covers the period 1996–2015. σ(M )/E(M ) is the market price of risk. t t 48

Figure 1: Model AALRRSV: Bayesian and distorted densities of x 60 Bayesian density Distorted density 50 40 30 20 10 0 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 x (LRR component) Notes: This figure plots Bayesian density and distorted density of the long-run risk component x for the AALRRSV model. The Bayesian density is x ∼N(xˆ ,ν ), and the distorted density is f˜(x |xˆ ,ν ,t). The t t t t t t distorteddensityisgeneratedfromsolvingthemodel. Thestatevectorisassumedtotakethevalue(xˆ =0, t ν = ν¯ (steady-state) and σ = µ ). Model parameters are set at posterior mean estimates presented in t t s Table 5. 49

Figure 2: Time-series of variables 0.4 0.2 s n ru 0 te r-0.2 -0.4 1941 1945 1949 1953 1957 1961 1965 1969 1973 1977 1981 1985 1989 1993 1997 2001 2005 2009 2013 e 0.05 ta r e e 0 rf k s-0.05 ir 1941 1945 1949 1953 1957 1961 1965 1969 1973 1977 1981 1985 1989 1993 1997 2001 2005 2009 2013 0.4 s n ru 0.2 te r s 0 s e-0.2 c x e-0.4 1941 1945 1949 1953 1957 1961 1965 1969 1973 1977 1981 1985 1989 1993 1997 2001 2005 2009 2013 h0.04 tw o rg 0.02 .s n 0 o c 1941 1945 1949 1953 1957 1961 1965 1969 1973 1977 1981 1985 1989 1993 1997 2001 2005 2009 2013 0.2 h tw o 0 rg .v id-0.2 1941 1945 1949 1953 1957 1961 1965 1969 1973 1977 1981 1985 1989 1993 1997 2001 2005 2009 2013 Year ThefigureshowsCRSPvalue-weightedindexreturns,one-yearTreasuryBillrates,excessreturns,per-capita log consumption growth, and log dividend growth rates for the 1941–2015 period. All series plotted are at an annual frequency and in real terms. Shaded areas represent NBER recessions. 50

Figure 3: Prior and Posterior Densities of Estimated Parameters of AAMS Model β γ ψ 0.96 0.98 1.00 −2 0 2 4 6 8 10 0.5 1.5 2.5 3.5 η p p l,l h,h 0 5 10 15 20 0.0 0.4 0.8 1.2 0.0 0.4 0.8 1.2 μ l μ h λ −0.10 −0.06 −0.02 0.02 −0.01 0.01 0.03 0.05 0 1 2 3 4 5 6 σ σ c d 0.00 0.02 0.04 0.06 0.00 0.10 0.20 This figure plots prior and posterior densities of the Ju and Miao (2012) model parameters. The solid lines depict posterior densities and dotted lines depict prior densities. The results are based on the U.S. annual data for 1941–2015. 51

Figure 4: Prior and Posterior Densities of Estimated Parameters of AAMSSV Model β γ ψ 0.96 0.98 1.00 −2 0 2 4 6 8 10 0.5 1.0 1.5 2.0 2.5 η pu pu l,l h,h 0 5 10 15 20 0.0 0.4 0.8 1.2 0.0 0.4 0.8 1.2 μ l μ h p l s ,l −0.10 −0.06 −0.02 0.02 0.00 0.02 0.04 0.5 0.7 0.9 1.1 p h s ,h σ l σ h 0.5 0.7 0.9 1.1 −0.01 0.01 0.03 0.00 0.02 0.04 0.06 λ σ d 0 1 2 3 4 5 6 0.00 0.10 0.20 This figure plots prior and posterior densities of the AAMSSV model, featuring ambiguity aversion, Markov switchinginbothconditionalmeanandvolatilityoftheconsumptionprocess. Thesolidlinesdepictposterior densitiesanddottedlinesdepictpriordensities. TheresultsarebasedontheU.S.annualdatafor1941–2015. 52

Figure 5: Prior and Posterior Densities of Estimated Parameters of AALRRSV Model β γ ψ 0.96 0.97 0.98 0.99 1.00 0 2 4 6 8 10 0.5 1.0 1.5 2.0 2.5 η μ ρ c x 0 20 40 60 0.012 0.016 0.020 0.024 0.0 0.5 1.0 φ λ φ x d 0.0 0.1 0.2 0.3 0.4 0 2 4 6 0 2 4 6 8 μ σ s w −0.01 0.01 0.03 −1e−04 2e−04 4e−04 6e−04 ThisfigureplotspriorandposteriordensitiesoftheAALRRSVmodel,featuringambiguityaversion,Kalman learning, stochastic volatility, and long-run risks in the conditional mean of the consumption process. The solid lines depict posterior densities and dotted lines depict prior densities. The results are based on the U.S. annual data for 1941–2015. 53

Figure 6: Prior and Posterior Densities of Estimated Parameters of EZMS Model β γ ψ 0.95 0.97 0.99 0 2 4 6 8 10 0.5 1.5 2.5 3.5 p p μ l,l h,h l 0.0 0.4 0.8 1.2 0.0 0.4 0.8 1.2 −0.10 −0.06 −0.02 μ h λ σ c 0.00 0.02 0.04 0.06 0 2 4 6 8 0.00 0.02 0.04 0.06 0.00 0.10 0.20 0.30 This figure plots prior and posterior densities of the EZMS model parameters, featuring Epstein and Zin preferencesandMarkovregimesintheconditionalmeanoftheconsumptiongrowthprocess. Thesolidlines depict posterior densities and dotted lines depict prior densities. The results are based on the U.S. annual data for 1941–2015. 54

Figure 7: Prior and Posterior Densities of Estimated Parameters of EZLRRSV BKY Model β γ ψ 0.96 0.98 1.00 0 5 10 15 0.5 1.0 1.5 2.0 2.5 3.0 μ ρ φ c x x 0.014 0.018 0.022 0.0 0.4 0.8 1.2 −0.1 0.0 0.1 0.2 0.3 0.4 λ φ φ d c 0 2 4 6 0 2 4 6 0 2 4 6 μ σ s w 0.00 0.02 0.04 0e+00 2e−04 4e−04 ThisfigureplotspriorandposteriordensitiesoftheEZLRRSVBKYmodelofBansaletal.(2012),featuring Epstein-Zin preferences, stochastic volatility and long-run risks in the conditional mean of the consumption process. The solid lines depict posterior densities and dotted lines depict prior densities. The results are based on the U.S. annual data for 1941–2015. 55

Figure 8: Impulse responses: AAMSSV Bayesian and distorted beliefs SDF 1 5 0.8 4 AAMS AAMSSV 0.6 3 Bayesian AAMS 0.4 Distorted AAMS 2 Bayesian AAMSSV 0.2 1 Distorted AAMSSV 0 0 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20 Time Time P/D Conditional equity premium 40 0.6 35 0.5 30 0.4 25 0.3 20 0.2 15 0.1 10 0 5 −0.1 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20 Time Time Conditional equity volatility VRP 0.6 50 0.5 40 0.4 30 0.3 20 0.2 10 0.1 0 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20 Time Time ThisfigureplotstheimpulseresponsefunctionsformodelsAAMSandAAMSSVwhenthemeanconsumption growthstateshiftsfrom µ toµ inthethirdperiod. Beforetherealizationoftheshock, meanconsumption h l growth is assumed to stay in state µ without the impact of innovation shocks. For the AAMSSV model, h thevolatilitystateisassumedtobeσ throughoutallperiods. Theresultsplottedareformodelparameters h set at posterior means of Bayesian MCMC estimates. 56

Figure 9: Impulse responses: AALRRSV LRR component and mean estimates SDF 0 3 2.5 −0.005 EZLRRSV AALRRSV 2 −0.01 LRR x 1.5 −0.015 Bayesian 1 Distorted −0.02 0.5 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20 Time Time P/D Conditional equity premium 26 0.055 24 0.05 22 20 0.045 18 0.04 16 0.035 14 12 0.03 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20 Time Time Conditional equity volatility VRP 0.17 2 0.165 1.5 0.16 0.155 1 0.15 0.145 0.5 0.14 0.135 0 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20 Time Time This figure plots the impulse response functions for models AALRRSV and EZLRRSV when a shock of size −4ϕ µ to x occurs in the third period. Before the realization of the shock, the AALRRSV economy is x s t assumed to stay in state (xˆ ,ν ,σ ) for which ∆c =µ ,∆d =µ ,x =0,σ =µ and ν =ν¯ (steady-state) t t t t c t d t t s t withouttheimpactofinnovationshocks. Thedistortedmeanestimateiscomputedbyapplyingtherejection sampling method and simulations. Before the realization of the shock, the EZLRRSV economy is assumed tostayinstate(x =0,σ2 =µ2)withouttheimpactofinnovationshocks. Theresultsplottedareformodel t t σ parameters set at posterior means of Bayesian MCMC estimates. 57

Figure 10: AAMS model: Conditional financial moments Equity premium Volatility of excess returns 0.8 0.8 0.7 0.6 0.6 0.4 0.5 0.4 0.2 0.3 0 0.2 Ju and Miao (2012) calibration -0.2 0.1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 π π t t Market price of risk VRP 4 150 3 100 2 50 1 0 0 -50 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 π π t t This figure plots conditional financial moments ranging from 5 to 95 percentile of simulated conditional moments for the AAMS model. The simulation is based on 12,000 Bayesian MCMC estimates of structural parameters. The dashed line plots the conditional moments calculated based on Ju and Miao’s calibration. 58

Figure 11: AAMSSV model: Conditional financial moments Equity premium Volatility of excess returns 1 1.4 1.2 0.8 1 0.6 0.8 0.4 0.6 0.2 0.4 0 0.2 -0.2 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 π π t t Market price of risk VRP 8 300 250 6 200 150 4 100 50 2 0 0 -50 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 π π t t This figure plots conditional financial moments ranging from 5 to 95 percentile of simulated conditional momentsfortheAAMSSVmodel. Thesimulationisbasedon12,000BayesianMCMCestimatesofstructural parameters. 59

Internet Appendix to “Does Smooth Ambiguity Matter for Asset Pricing?”

1 Numerical methods WeusethecollocationprojectionmethodwithChebyshevpolynomialstosolveassetpricingmodels in the paper. See Judd (1992) for an introduction to projection methods and Pohl et al. (2017) for applications to solving models with long-run risks. We solve each model in two steps. In the first step, we use the projection method to solve the functional equation for the value function V (C) to obtain the wealth-consumption ratio. Suppose t that the vector of state variables for a model is denoted by z (e.g., z = {π } in model AAMS). t t t By homogeneity, we have V (C) = C G(z ) where G(z ) is a function to be determined. As shown t t t t by Epstein and Zin (1989), the wealth-consumption ratio W /C is given by t t W t 1 (cid:18) V t (cid:19)1− ψ 1 = . C 1−β C t t In the second step, we apply the projection method to solve the Euler equation to obtain the pricedividend ratio, given that we can determine the SDF M from the solution in the first step. We t,t+1 denote the current state of the economy by z and the next period’s state by z(cid:48). 1.1 Solving the AAMS Model This model is developed by Ju and Miao (2012). See “Ambiguity, Learning, and Asset Returns: Technical Appendix” for details about the numerical method. The functional equation for G(π) implied by the generalized recursive smooth ambiguity utility function is given by   1 G(π) = (1−β)+β (cid:18) E (cid:20) (cid:16) E (cid:104) G (cid:0) π(cid:48)(cid:1)1−γ exp (cid:0) (1−γ)∆c (cid:0) s(cid:48)(cid:1)(cid:1) (cid:12) (cid:12) (cid:12) s(cid:48) (cid:105)(cid:17) 1 1 − − γ η(cid:12) (cid:12) (cid:12)π (cid:21)(cid:19)1− 1− 1/ η ψ  1−1/ψ . (1) (cid:12) 1

The intertemporal marginal rate of substitution (or stochastic discount factor) is given by M (cid:0) π(cid:48),s(cid:48)|π (cid:1) = βexp (cid:18) − 1 ∆c (cid:0) s(cid:48)(cid:1) (cid:19)(cid:18) G(π(cid:48))exp(∆c(s(cid:48))) (cid:19) ψ 1−γ ψ R(G(π(cid:48))exp(∆c(s(cid:48)))|π) (cid:16) (cid:104) (cid:12) (cid:105)(cid:17) 1 −(η−γ) E G(π(cid:48))1−γexp((1−γ)∆c(s(cid:48)))(cid:12)s(cid:48),π 1−γ (cid:12) ×  .  R(G(π(cid:48))exp(∆c(s(cid:48)))|π)  The price-dividend ratio ϕ(π) satisfies the Euler equation ϕ(π) = E(cid:2) M (cid:0) π(cid:48),s(cid:48)|π (cid:1)(cid:0) 1+ϕ (cid:0) π(cid:48)(cid:1)(cid:1) exp (cid:0) ∆d (cid:0) s(cid:48)(cid:1)(cid:1)(cid:12) (cid:12)π (cid:3) . (2) The laws of motion of consumption and dividend growth are ∆c(s) = µ(s)+σ (cid:15) , (cid:15) ∼ N (0,1) c c c ∆d(s) = λ∆c(s)+g +σ˜ (cid:15) , (cid:15) ∼ N (0,1) d d d d where the transition probabilities are Pr (cid:0) s(cid:48) = l|s = l (cid:1) = p , Pr (cid:0) s(cid:48) = h|s = h (cid:1) = p ll hh and (cid:15) and (cid:15) are two independent innovation shocks. c d The (nonlinear) law of motion of the state variable π is p f(∆c(s(cid:48))|s(cid:48) = h)π+(1−p )f(∆c(s(cid:48))|s(cid:48) = l)(1−π) π(cid:48) = hh ll . f(∆c(s(cid:48))|s(cid:48) = h)π+f(∆c(s(cid:48))|s(cid:48) = l)(1−π) We approximate the solution functions G(π) and ϕ(π) by Chebyshev polynomials, namely, Gˆ(cid:0) π;aG(cid:1) = (cid:88) nπ aGT (t ), ϕˆ(π;aϕ) = (cid:88) nπ aϕT (t ) k k π k k π k=0 k=0 where T : [−1,1] → R, k = 0,1,...,n are Chebyshev polynomials and the transformation of the k π 2

argument for the polynomial is given by (cid:18) (cid:19) π−π min t = 2 −1 π π −π max min with π = 0 and π = 1. To implement the collocation method, we solve the two functional min max equations (1) and (2) on a grid of π obtained by applying the inverse of the transformation to the n +1 zeros of the Chebyshev polynomial T . π nπ+1 Equations (1) and (2) define two residual functions that are to be minimized sequentially by choosing the coefficients aG and aϕ. The collocation projection method leads to two square systems of nonlinear equations, which can be solved with a nonlinear equations solver (e.g., Powell’s hybrid algorithm). Because the underlying innovation shocks are Gaussian, we use Gauss-Hermite quadrature to calculate conditional expectations in the residual functions. 1.2 Solving the AAMSSV Model Compared to AAMS, the AAMSSV model has one additional state variable sσ indicating the t volatility state. It follows that G(π,sσ) satisfies the equation   1 G(π,sσ) = (1−β)+β (cid:18) E (cid:20) (cid:16) E (cid:104) G (cid:0) π(cid:48),sσ(cid:48)(cid:1)1−γ exp (cid:0) (1−γ)∆c (cid:0) sµ(cid:48),sσ(cid:48)(cid:1)(cid:1) (cid:12) (cid:12) (cid:12) sµ(cid:48),sσ (cid:105)(cid:17) 1 1 − − γ η(cid:12) (cid:12) (cid:12)π (cid:21)(cid:19)1− 1− 1/ η ψ  1−1/ψ . (cid:12) The SDF and Euler equation are given by M (cid:0) π(cid:48),sσ(cid:48),sµ(cid:48)|π,sσ(cid:1) = βexp (cid:18) − 1 ∆c (cid:0) sµ(cid:48),sσ(cid:48)(cid:1) (cid:19)(cid:18) G(π(cid:48),sσ(cid:48))exp(∆c(sµ(cid:48),sσ(cid:48))) (cid:19) ψ 1−γ ψ R(G(π(cid:48),sσ(cid:48))exp(∆c(sµ(cid:48),sσ(cid:48)))|π,sσ) (cid:16) (cid:104) (cid:12) (cid:105)(cid:17) 1 −(η−γ) E G(π(cid:48),sσ(cid:48))1−γexp((1−γ)∆c(sµ(cid:48),sσ(cid:48)))(cid:12)sµ(cid:48),π,sσ 1−γ (cid:12) ×   R(G(π(cid:48),sσ(cid:48))exp(∆c(sµ(cid:48),sσ(cid:48)))|π,sσ)  and ϕ(π,sσ) = E(cid:2) M (cid:0) π(cid:48),sσ(cid:48),sµ(cid:48)|π,sσ(cid:1)(cid:0) 1+ϕ (cid:0) π(cid:48),sσ(cid:48)(cid:1)(cid:1) exp (cid:0) ∆d (cid:0) sµ(cid:48),sσ(cid:48)(cid:1)(cid:1)(cid:12) (cid:12)π,sσ(cid:3) . 3

The laws of motions for consumption and dividend growth are ∆c(sµ,sσ) = µ(sµ)+σ(sσ)(cid:15) , (cid:15) ∼ N (0,1) c c ∆d(sµ,sσ) = λ∆c(sµ,sσ)+g +σ˜ (cid:15) , (cid:15) ∼ N (0,1) d d d d where the transition probabilities for the two independent Markov chains of sµ and sσ are given by Pr (cid:0) sσ(cid:48) = l|sσ = l (cid:1) = pσ, Pr (cid:0) sσ(cid:48) = h|sσ = h (cid:1) = pσ ll hh Pr (cid:0) sµ(cid:48) = l|sµ = l (cid:1) = pµ, Pr (cid:0) sµ(cid:48) = h|sµ = h (cid:1) = pµ ll hh The law of motion of the state variable π is given by the Bayes’ rule pµ f(∆c(sµ(cid:48),sσ(cid:48))|sµ(cid:48) = h,sσ(cid:48))π+ (cid:0) 1−pµ(cid:1) f(∆c(sµ(cid:48),sσ(cid:48))|sµ(cid:48) = l,sσ(cid:48))(1−π) π(cid:48) = hh ll f(∆c(sµ(cid:48),sσ(cid:48))|sµ(cid:48) = h,sσ(cid:48))π+f(∆c(sµ(cid:48),sσ(cid:48))|sµ(cid:48) = l,sσ(cid:48))(1−π) We approximate the solutions to G(π,sσ) and ϕ(π,sσ) by Chebyshev polynomials as Gˆ(cid:0) π,sσ = l;aG(cid:1) = (cid:88) nπ aG T (t ), Gˆ(cid:0) π,sσ = h;aG(cid:1) = (cid:88) nπ aG T (t ) l k,l k π h k,h k π k=0 k=0 ϕˆ (cid:0) π,sσ = l;aϕ(cid:1) = (cid:88) nπ aϕ T (t ), ϕˆ (cid:0) π,sσ = h;aϕ(cid:1) = (cid:88) nπ aϕ T (t ) l k,l k π h k,h k π k=0 k=0 i.e., we seek four sets of coefficients (cid:0) aG,aG,aϕ,aϕ(cid:1) that minimize the residual functions. l h l h 1.3 Solving the AALRRSV Model We consider the long-run risk model ∆c = µ +x +σ (cid:15) t+1 c t+1 t c,t+1 ∆d = µ +λx +ϕ σ (cid:15) t+1 d t+1 d t d,t+1 x = ρ x +ϕ σ (cid:15) t+1 x t x t x,t+1 σ2 = µ2 +ρ (cid:0) σ2−µ2(cid:1) +σ (cid:15) t+1 σ s t s w w,t+1 (cid:15) ,(cid:15) ,(cid:15) ,(cid:15) ∼ i.i.d.N (0,1). c,t+1 d,t+1 x,t+1 w,t+1 4

where the long-run risk component x is unobservable. We define xˆ = E[x |I ] and ν = t t+1|t t+1 t t+1|t (cid:104) (cid:105) (cid:0) (cid:1)2 E x −xˆ |I where I denotes available information at time t. It immediately follows t+1 t+1|t t t that xˆ = ρ xˆ , and ν = ρ2ν +ϕ2σ2. t+1|t x t t+1|t x t x t The Kalman filter implies the following updating equations   (cid:20) (cid:21) vc xˆ t+1 = xˆ t+1|t +ν t+1|t 1 λ F t − + 1 1|t   t+1|t   vd t+1|t (cid:20) (cid:21) (cid:20) (cid:21)(cid:48) ν t+1 = ν t+1|t −ν t 2 +1|t 1 λ F t − + 1 1|t 1 λ where F is given by t+1|t   ν +σ2 λν F =  t+1|t t t+1|t  t+1|t   λν λ2ν +ϕ2σ2 t+1|t t+1|t d t (cid:20) (cid:21) and the innovation vector vc vd is given by t+1|t t+1|t     vc ∆c −µ −ρ xˆ  t+1|t  =  t+1 c x t .     vd ∆d −µ −λρ xˆ t+1|t t+1 d x t Expressed as an intertemporal equation, the solution function G(xˆ,ν,σ) satisfies G(xˆ,ν,σ) =  (1−β)+β (cid:18) E (cid:20) (cid:16) E (cid:104) G (cid:0) xˆ(cid:48),ν(cid:48),σ(cid:48)(cid:1)1−γ exp (cid:0) (1−γ)∆c (cid:0) x(cid:48)(cid:1)(cid:1) (cid:12) (cid:12)x,σ (cid:105)(cid:17) 1 1 − − γ η(cid:12) (cid:12) (cid:12)xˆ,ν (cid:21)(cid:19) 1 1 − − ψ η 1   1− 1 ψ 1 .  (cid:12) (cid:12)  5

The SDF and Euler equation are given by M (cid:0) x(cid:48),x,xˆ(cid:48),ν(cid:48),σ(cid:48)|xˆ,ν,σ (cid:1) = βexp (cid:18) − 1 ∆c (cid:0) x(cid:48)(cid:1) (cid:19)(cid:18) G(xˆ(cid:48),ν(cid:48),σ(cid:48))exp(∆c(x(cid:48))) (cid:19) ψ 1−γ ψ R(G(xˆ(cid:48),ν(cid:48),σ(cid:48))exp(∆c(x(cid:48)))|xˆ,ν,σ) (cid:16) (cid:104) (cid:12) (cid:105)(cid:17) 1 −(η−γ) E G(xˆ(cid:48),ν(cid:48),σ(cid:48))1−γexp((1−γ)∆c(x(cid:48)))(cid:12)x,xˆ,ν,σ 1−γ (cid:12) ×   R(G(xˆ(cid:48),ν(cid:48),σ(cid:48))exp(∆c(x(cid:48)))|xˆ,ν,σ)  and ϕ(xˆ,ν,σ) = E(cid:2) M (cid:0) x(cid:48),x,xˆ(cid:48),ν(cid:48),σ(cid:48)|xˆ,ν,σ (cid:1)(cid:0) 1+ϕ (cid:0) xˆ(cid:48),ν(cid:48),σ(cid:48)(cid:1)(cid:1) exp (cid:0) ∆d (cid:0) x(cid:48)(cid:1)(cid:1)(cid:12) (cid:12)xˆ,ν,σ (cid:3) . We approximate the solution functions G(xˆ,ν,σ) and ϕ(xˆ,ν,σ) by three-dimensional product Chebyshev polynomials, namely, Gˆ(cid:0) xˆ,ν,σ;aG(cid:1) = (cid:88) n xˆ (cid:88) nν (cid:88) nσ aGaGaG T (t )T (t )T (t ) k xˆ kν kσ k xˆ xˆ kν ν kσ σ k xˆ =0kν=0kσ=0 ϕˆ(xˆ,ν,σ;aϕ) = (cid:88) n xˆ (cid:88) nν (cid:88) nσ aϕ aϕ aϕ T (t )T (t )T (t ). k xˆ kν kσ k xˆ xˆ kν ν kσ σ k xˆ =0kν=0kσ=0 In constructing Chebyshev polynomials as basis functions, we obtain the lower and upper bounds for each state variable by simulations. Because x ∼ N (xˆ ,ν ), we use Gauss-Hermite quadrature t t t to compute the conditional expectation involving state x . To compute conditional expectations t with respect to the underlying shocks ((cid:15) ,(cid:15) ,(cid:15) ,(cid:15) ), we apply the monomial method with degree c d x σ 5, see Judd (1999) for details of the monomial method. If the dimension of underlying shocks is d, the monomial method requires 2d2 + 1 points to compute an expectation, whereas Gauss- Hermite quadrature requires Nd nodes with N being the number of nodes in one dimension. When the dimension of underlying shocks is large, the monomial method is much more efficient than quadrature methods. This gain in efficiency is particularly important for our structural estimation. A number of simulations suggest that for our model the monomial method yields accurate results compared with Gauss-Hermite quadrature. To implement the collocation method, we solve the two square systems of nonlinear equations derived from equilibrium conditions on a grid of dimension (n +1)×(n +1)×(n +1) for the xˆ ν σ state variables. The grid is constructed from zeros of Chebyshev polynomials of all state variables. 6

An alternative approach is to discretize the AR(1) process of σ2 into a n−state Markov chain t by the method developed in Tauchen (1986). Caldara et al. (2012) adopt this approach to solve DSGE models with recursive preferences and stochastic volatility. To avoid negative volatility states in the Markov chain, we keep positive values only and normalize transition probabilities accordingly. Assuch,giveneachvolatilitystateσ ,thesolutionfunctionsG(xˆ,ν,σ )andϕ(xˆ,ν,σ ) i i i can be approximated by two-dimensional product Chebyshev polynomials in xˆ and ν. Through simulations, we find that this approach yields results that are close to the approximation with three-dimensional product Chebyshev polynomials. 1.4 Solving the EZLRRSV Model The laws of motion of ∆c, ∆d, x and σ are given by the long-run risk model ∆c = µ +x+σ(cid:15) c c ∆d = µ +λx+ϕ σ(cid:15) +ϕ σ(cid:15) d d d c c x(cid:48) = ρ x+ϕ σ(cid:15) x x x σ2(cid:48) = µ2 +ρ (cid:0) σ2−µ2(cid:1) +σ (cid:15) σ σ σ w w (cid:15) ,(cid:15) ,(cid:15) ,(cid:15) ∼ i.i.d.N (0,1). c d x w The solution function G (cid:0) x,σ2(cid:1) satisfies  (cid:16) (cid:104) (cid:12) (cid:105)(cid:17) 1− ψ 1  1− 1 ψ 1 G (cid:0) x,σ2(cid:1) = (1−β)+β E G (cid:0) x(cid:48),σ2(cid:48)(cid:1)1−γ exp((1−γ)∆c)(cid:12) (cid:12) x,σ2 1−γ  The SDF and Euler equation are given by M (cid:0) x(cid:48),σ2(cid:48)|x,σ2(cid:1) = βexp (cid:18) − 1 ∆c (cid:19) (cid:32) G (cid:0) x(cid:48),σ2(cid:48)(cid:1) exp(∆c) (cid:33) ψ 1−γ ψ R(G(x(cid:48),σ2(cid:48))exp(∆c)|x,σ2) and ϕ (cid:0) x,σ2(cid:1) = E(cid:2) M (cid:0) x(cid:48),σ2(cid:48)|x,σ2(cid:1)(cid:0) 1+ϕ (cid:0) x,σ2(cid:1)(cid:1) exp(∆d) (cid:12) (cid:12)x,σ2(cid:3) 7

We approximate the solution functions G (cid:0) x,σ2(cid:1) and ϕ (cid:0) x,σ2(cid:1) by two-dimensional product Chebyshev polynomials in x and σ2: Gˆ(cid:0) x,σ2;aG(cid:1) = (cid:88) nx (cid:88) nσ aGaG T (t )T (t ) kx kσ kx x kσ σ kx=0kσ=0 ϕˆ (cid:0) x,σ2;aϕ(cid:1) = (cid:88) n xˆ (cid:88) nσ aϕ aϕ T (t )T (t ). kx kσ kx x kσ σ kx=0kσ=0 2 Numerical accuracy of the solution method WeusethemethodproposedbyJudd(1992)toassessnumericalaccuracyofournumericalsolutions. ThenumericalaccuracycheckisthroughcomputingtheEulerequationerror. Previousstudiessuch as Guerrieri and Iacoviello (2015) and Collard, Mukerji, Sheppard, and Tallon (2017) rely on this approach to assess the accuracy of their numerical solutions. Note that instead of computing the Euler equation error implied by calibrated parameters as previous studies do, we compute the error based on the MCMC chain of parameter estimates for each asset pricing model. For each model, we compute several metrics of the error on a chain of estimates (12,000 sets of estimates) obtained from the GSM Bayesian estimation. FortheAAMSmodel, theEulerequationerrorsdefinedonthedividendclaimandconsumption claim are respectively given by    −(η−γ) −ψ −C t +  E t   βC t − + 1 1 /ψ (cid:16) Rt V ( t V + t+ 1 1) (cid:17)1/ψ−γ  (cid:16) E {st+1 R ,t t } ( [ V V t t + 1 + − 1 1 ) γ] (cid:17) 1− 1 γ  D Pt t + + P 1 1 t +1 D D t+ t 1    Dt EulerErrD = t C t    −(η−γ) −ψ −C t +  E t   βC t − + 1 1 /ψ (cid:16) Rt V ( t V + t+ 1 1) (cid:17)1/ψ−γ  (cid:16) E {st+1 R ,t t } ( [ V V t t + 1 + − 1 1 ) γ] (cid:17) 1− 1 γ  W W Ct t t + + − 1 1 1 C C t+ t 1    Ct EulerErrC = . t C t TheerrorsaredefinedinasimilarwayforothermodelsincludingAAMSSV,AALRRSV,EZLRRSV and EZMS. The differences are only with regard to the SDF and conditioning state variables. This measure is expressed as a fraction of consumption goods, namely the residual of the Euler equation normalizedbyconsumption. EulerErrD (EulerErrC)quantifiestheerrortheagentwouldcommit t t 8

if he use the approximate solution for the price of the dividend (consumption) claim to decide on marginal investment. Following Judd (1992), we consider several metrics of the error to evaluate numerical accuracy: ED = log (cid:0)E(cid:2)(cid:12) (cid:12)EulerErrD(cid:12) (cid:12) (cid:3)(cid:1) , ED = log (cid:16) E (cid:104) (cid:0) EulerErrD(cid:1)2 (cid:105)(cid:17) 1 10 t 2 10 t EC = log (cid:0)E(cid:2)(cid:12) (cid:12)EulerErrC(cid:12) (cid:12) (cid:3)(cid:1) , EC = log (cid:16) E (cid:104) (cid:0) EulerErrC(cid:1)2 (cid:105)(cid:17) . 1 10 t 2 10 t We report the mean, 5 percentile and 95 percentile of each metric evaluated on the MCMC chains ofestimates. ItisimportanttonotethatwecomputetheEulerequationerroroutsidethegridthat we use to implement the collocation projection method. This is done because we want to assess whether our approximate solutions perform well for simulated data under each model, and because in the GSM Bayesian estimation we use the simulated data to find the mapping recovery from structural parameters to the auxiliary model parameters. We report all measures in log terms. 10 For example, a value of ED equal to -3 suggests that if an agent relies on the approximate solution 1 of the price of the dividend claim, he would expect to make a mistake of 1 dollar for each $1000 risky investment. The economic interpretation is similar for EC. The metric E D(C) measures the 1 2 quadratic average of the error. Results reported in Table 1 show that our approximate solutions are accurate. 9

Table 1: Numerical accuracy: Euler errors Model ED ED EC EC 1 2 1 2 AAMS Mean -2.654 -5.281 -3.656 -7.282 95 percentile -2.179 -4.334 -3.230 -6.437 5 percentile -3.256 -6.480 -4.233 -8.420 AAMSSV Mean -2.594 -4.940 -4.100 -7.985 95 percentile -1.826 -3.282 -3.165 -6.120 5 percentile -3.918 -7.679 -5.612 -11.072 AALRRSV Mean -2.207 -4.083 -4.093 -7.550 95 percentile -1.633 -2.956 -2.983 -5.551 5 percentile -2.626 -4.951 -4.932 -9.297 EZLRRSV Mean -2.877 -5.387 -2.820 -5.255 95 percentile -2.724 -5.066 -2.690 -4.966 5 percentile -3.126 -5.880 -3.044 -5.708 EZMS Mean -3.751 -7.335 -4.572 -8.987 95 percentile -3.251 -6.026 -3.979 -7.834 5 percentile -4.621 -9.077 -5.444 -10.737 10

Table 2: Prior: AAMS Parameter Min Max µ σ β 0.9 0.995 0.985 0.005 γ 0.1 100 5 1 ψ 0.1 10 1.5 0.2 η γ 200 8.87 2 p 0.2 0.999 0.516 0.13 ll p 0.2 0.999 0.978 0.24 hh µ -0.08 0.00 -0.0678 0.017 l µ 0.00 0.08 0.022 0.006 h λ 1 6 2.74 0.8 σ 0.004 0.06 0.03 0.0075 c σ 0.03 0.3 0.13 0.03 d 3 Priors on structural parameters We report support conditions (Min and Max), prior location and scale parameters for structural parameters in models AAMS, AAMSSV, AALRRSV and EZLRRSV.1 For each model, the prior is the combination of the product of independent normal density functions and support conditions. The product of independent normal density functions is given by n˜ ξ(θ) = (cid:89) N (cid:2) θ | (cid:0) θ∗,σ2(cid:1)(cid:3) i i θ i=1 wheren˜ denotesthenumberofparameters. Becausethispriorisintersectedwithsupportconditions that are not all of product form, and because a support condition that rejects parameter values in the MCMC chain implies extreme parameter values such that the solution method fails, this is not an independence prior. 1 TheEZMSmodelhasthesameprioronparametersastheAAMSmodeldoesexceptfortheabsenceoftheambiguity aversion parameter η. 11

Table 3: Prior: AAMSSV Parameter Min Max µ σ β 0.9 0.995 0.985 0.005 γ 0.1 100 5 1 ψ 0.1 10 1.5 0.2 η γ 200 8.87 2 p 0.2 0.999 0.516 0.13 ll p 0.2 0.999 0.978 0.24 hh µ -0.08 0.00 -0.0678 0.017 l µ 0.00 0.08 0.022 0.006 h pσ 0.2 0.999 0.85 0.07 ll pσ 0.2 0.999 0.85 0.07 hh σ 0.004 0.06 0.015 0.0038 l σ 0.004 0.06 0.03 0.0075 h λ 1 6 2.74 0.8 σ 0.03 0.3 0.13 0.03 d Table 4: Prior: AALRRSV Parameter Min Max µ σ β 0.9 0.995 0.985 0.005 γ 0.1 100 5 1 ψ 0.1 10 1.5 0.2 η γ 200 25 5 µ 0.012 0.025 0.02 0.001 c ρ -0.99 0.99 0.8 0.2 x φ 0.01 0.5 0.15 0.04 x λ 1 10 3 0.8 φ 0.5 10 3 0.8 d µ 0.001 0.1 0.02 0.005 s ρ 0.3 0.99 0.8 0.2 s σ 1e-5 0.001 2.5e-4 6.25e-5 w 12

Table 5: Prior: EZLRRSV Parameter Min Max µ σ β 0.9 0.995 0.985 0.005 γ 0.1 100 5 1 ψ 0.1 10 1.5 0.2 µ 0.012 0.025 0.019 0.001 c ρ -0.99 0.99 0.80 0.20 x φ 0.01 0.5 0.15 0.04 x λ 1 10 3 0.8 φ 0.5 10 3 0.8 d φ 1 10 3 0.8 c µ 0.001 0.10 0.02 0.005 s ρ 0.30 0.99 0.8 0.2 s σ 1e-5 0.001 2.5e-4 6.25e-5 w 4 GSM estimation results with augmented priors WealsoperformtheGSMBayesianestimationwithaugmentedpriorstakingintoaccountmoments of asset returns and consumption and dividend growth. The aim of this estimation is to examine whether our GSM estimation results reported in the paper are robust to the augmented priors. The augmented prior on moments is specified to be the product of independent normal density functions as n¯ ξ¯(m) = (cid:89) N (cid:2) m | (cid:0) m∗,σ2 (cid:1)(cid:3) k k m k k=1 wherem ≡ (m ,m ,...,m )isavectorofmomentsunderconsideration. Thelocationandscalepa- 1 2 n¯ rameters for the moment m are m∗ and σ respectively. We use the following location parameter k k m k values for eight moments to form the prior. E(rf) = 0.014, σ(rf) = 0.028, E(r ) = 0.068, σ(r ) = 0.18 t t t t E(∆c ) = 0.018, σ(∆c ) = 0.021, E(∆d ) = 0.018, σ(∆d ) = 0.14 t t t t Thescaleparametersaresetatvaluessuchthatthepriorput95%ofitsmassonbeingwithin10%of itslocationparameter. WesimulatethesemomentsfromassetpricingmodelsintheGSMBayesian estimation. The results reported below show that the GSM estimation with the augmented priors 13

yield similar results to those reported in the paper. 14

Table 6: GSM Estimation Results: the AAMS Model Prior Posterior Parameter Mean Median 5% 95% Mean Median 5% 95% β 0.991 0.991 0.989 0.991 0.980 0.980 0.977 0.983 γ 4.847 4.750 2.750 6.750 2.219 2.281 0.766 3.531 ψ 0.616 0.563 0.563 0.813 2.202 2.180 1.836 2.680 η 13.155 13.500 11.500 13.500 5.557 5.219 4.594 7.922 p 0.405 0.406 0.406 0.406 0.860 0.866 0.764 0.923 ll p 0.812 0.813 0.813 0.813 0.997 0.997 0.996 0.997 hh µ -0.043 -0.043 -0.043 -0.043 -0.048 -0.048 -0.054 -0.041 l µ 0.033 0.033 0.033 0.033 0.020 0.020 0.018 0.021 h λ 2.803 2.750 2.750 3.250 2.791 2.734 2.391 3.547 σ 0.006 0.006 0.006 0.006 0.020 0.020 0.018 0.024 c σ 0.130 0.133 0.117 0.133 0.135 0.136 0.124 0.146 d MCMC repetitions 10,000 12,000 This table presents prior and posterior marginal means, medians, 5 and 95 percentiles of model parameters for the AAMS model. The GSM estimation imposes the augmented prior on moments of asset returns and consumptionanddividendgrowth. MCMCrepetitionsaftertransientshavedissipatedarereportedforboth the prior and posterior. Estimation results are for annual data 1941–2015. 15

Table 7: GSM Estimation Results: the AAMSSV Model Prior Posterior Parameter Mean Median 5% 95% Mean Median 5% 95% β 0.989 0.989 0.989 0.989 0.978 0.978 0.974 0.982 γ 4.982 5.250 3.250 6.250 0.848 0.844 0.219 1.531 ψ 0.450 0.438 0.438 0.438 1.779 1.715 1.434 2.199 η 9.496 9.500 9.500 9.500 10.232 9.281 8.531 15.500 pµ 0.282 0.281 0.281 0.281 0.706 0.728 0.611 0.774 ll pµ 0.812 0.813 0.813 0.813 0.998 0.999 0.997 0.999 hh µ -0.066 -0.066 -0.066 -0.066 -0.055 -0.054 -0.060 -0.050 l µ 0.033 0.033 0.033 0.033 0.018 0.018 0.016 0.019 h pσ 0.863 0.859 0.734 0.984 0.989 0.989 0.982 0.993 ll pσ 0.840 0.859 0.703 0.953 0.989 0.990 0.984 0.993 hh σ 0.006 0.005 0.005 0.009 0.013 0.013 0.006 0.021 l σ 0.006 0.006 0.006 0.010 0.029 0.029 0.026 0.032 h λ 3.064 3.250 2.750 3.250 2.570 2.547 1.984 3.172 σ 0.107 0.102 0.086 0.117 0.134 0.134 0.122 0.146 d MCMC repetitions 10,000 12,000 This table presents prior and posterior marginal means, medians, 5 and 95 percentiles of model parameters for the AAMSSV model. The GSM estimation imposes the augmented prior on moments of asset returns and consumption and dividend growth. MCMC repetitions after transients have dissipated are reported for both the prior and posterior. Estimation results are for annual data 1941–2015. 16

ledoM VSRRLAA eht :stluseR noitamitsE MSG :8 elbaT roiretsoP roirP %59 %5 naideM naeM %59 %5 naideM naeM retemaraP 599.0 099.0 299.0 299.0 399.0 589.0 989.0 989.0 β 609.8 651.7 130.8 310.8 578.01 521.7 578.8 769.8 γ 759.0 558.0 598.0 998.0 604.1 609.0 651.1 551.1 ψ 005.14 005.91 005.13 580.13 000.14 000.52 000.33 521.33 η 220.0 910.0 020.0 020.0 020.0 810.0 910.0 910.0 µ c 359.0 729.0 049.0 049.0 448.0 448.0 448.0 448.0 ρ x 332.0 181.0 012.0 902.0 763.0 372.0 503.0 113.0 φ x 302.3 354.2 437.2 487.2 578.3 578.2 521.3 932.3 λ 953.5 906.4 489.4 599.4 573.5 573.4 578.4 009.4 φ d 020.0 910.0 020.0 020.0 110.0 010.0 010.0 110.0 µ s 059.0 059.0 059.0 059.0 969.0 969.0 969.0 969.0 ρ s 40-E81.2 40-E39.1 40-E90.2 40-E80.2 40-E89.1 40-E86.1 40-E86.1 40-E48.1 σ w 000,21 000,01 snoititeper CMCM MSG ehT .ledom VSRRLAA eht rof sretemarap ledom fo selitnecrep 59 dna 5 ,snaidem ,snaem lanigram roiretsop dna roirp stneserp elbat sihT stneisnart retfa snoititeper CMCM .htworg dnedivid dna noitpmusnoc dna snruter tessa fo stnemom no roirp detnemgua eht sesopmi noitamitse .5102–1491 atad launna rof era stluser noitamitsE .roiretsop dna roirp eht htob rof detroper era detapissid evah 17

Table 9: GSM Estimation Results: the EZMS Model Prior Posterior Parameter Mean Median 5% 95% Mean Median 5% 95% β 0.986 0.985 0.985 0.987 0.981 0.981 0.978 0.984 γ 10.221 10.250 10.250 10.250 4.013 3.953 3.469 4.672 ψ 0.421 0.438 0.313 0.438 2.397 2.359 1.859 2.984 p 0.343 0.344 0.344 0.344 0.904 0.897 0.860 0.951 ll p 0.812 0.813 0.813 0.813 0.996 0.997 0.992 0.997 hh µ -0.059 -0.059 -0.059 -0.059 -0.039 -0.041 -0.052 -0.018 l µ 0.029 0.029 0.029 0.029 0.022 0.021 0.018 0.029 h λ 3.252 3.250 3.250 3.250 3.192 3.172 2.641 3.766 σ 0.006 0.006 0.006 0.006 0.019 0.019 0.016 0.023 c σ 0.105 0.102 0.102 0.117 0.132 0.133 0.117 0.145 d MCMC repetitions 10,000 12,000 This table presents prior and posterior marginal means, medians, 5 and 95 percentiles of model parameters for the EZMS model. The GSM estimation imposes the augmented prior on moments of asset returns and consumptionanddividendgrowth. MCMCrepetitionsaftertransientshavedissipatedarereportedforboth the prior and posterior. Estimation results are for annual data 1941–2015. 18

ledoM VSRRLZE eht :stluseR noitamitsE MSG :01 elbaT roiretsoP roirP %59 %5 naideM naeM %59 %5 naideM naeM retemaraP 689.0 189.0 489.0 489.0 399.0 989.0 399.0 399.0 β 834.01 839.6 834.8 245.8 057.7 052.5 057.6 008.6 γ 320.3 135.2 987.2 377.2 886.1 313.1 313.1 573.1 ψ 420.0 120.0 320.0 220.0 120.0 910.0 020.0 020.0 µ c 099.0 179.0 989.0 589.0 839.0 839.0 839.0 739.0 ρ x 801.0 060.0 170.0 770.0 633.0 591.0 503.0 092.0 φ x 604.5 886.3 964.4 815.4 057.3 052.2 052.2 285.2 λ 839.4 651.4 135.4 645.4 057.5 057.3 057.4 247.4 φ d 656.1 610.1 651.1 712.1 052.5 057.2 052.4 200.4 φ c 020.0 910.0 910.0 910.0 710.0 110.0 110.0 310.0 µ s 059.0 059.0 059.0 059.0 839.0 839.0 839.0 839.0 ρ s 40-E30.2 40-E78.1 40-E39.1 40-E39.1 40-E86.1 40-E73.1 40-E73.1 40-E05.1 σ w 000,21 000,01 snoititeper CMCM MSG ehT .ledom VSRRLZE eht rof sretemarap ledom fo selitnecrep 59 dna 5 ,snaidem ,snaem lanigram roiretsop dna roirp stneserp elbat sihT stneisnart retfa snoititeper CMCM .htworg dnedivid dna noitpmusnoc dna snruter tessa fo stnemom no roirp detnemgua eht sesopmi noitamitse .5102–1491 atad launna rof era stluser noitamitsE .roiretsop dna roirp eht htob rof detroper era detapissid evah 19

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